The [[wp:Sieve_of_Eratosthenes|Sieve of Eratosthenes]] is a simple algorithm that finds the prime numbers up to a given integer.
Task
Implement the Sieve of Eratosthenes algorithm, with the only allowed optimization that the outer loop can stop at the square root of the limit, and the inner loop may start at the square of the prime just found.
That means especially that you shouldn't optimize by using pre-computed ''wheels'', i.e. don't assume you need only to cross out odd numbers (wheel based on 2), numbers equal to 1 or 5 modulo 6 (wheel based on 2 and 3), or similar wheels based on low primes.
If there's an easy way to add such a wheel based optimization, implement it as an alternative version.
;Note:
- It is important that the sieve algorithm be the actual algorithm used to find prime numbers for the task.
Related tasks
- [[Emirp primes]]
- [[count in factors]]
- [[prime decomposition]]
- [[factors of an integer]]
- [[extensible prime generator]]
- [[primality by trial division]]
- [[factors of a Mersenne number]]
- [[trial factoring of a Mersenne number]]
- [[partition an integer X into N primes]]
- [[sequence of primes by Trial Division]]
360 Assembly
For maximum compatibility, this program uses only the basic instruction set. <lang 360_Assembly>* Sieve of Eratosthenes ERATOST CSECT USING ERATOST,R12 SAVEAREA B STM-SAVEAREA(R15) DC 17F'0' DC CL8'ERATOST' STM STM R14,R12,12(R13) save calling context ST R13,4(R15) ST R15,8(R13) LR R12,R15 set addessability
---- CODE LA R4,1 I=1 LA R6,1 increment L R7,N limit
LOOPI BXH R4,R6,ENDLOOPI do I=2 to N LR R1,R4 R1=I BCTR R1,0 LA R14,CRIBLE(R1) CLI 0(R14),X'01' BNE ENDIF if not CRIBLE(I) LR R5,R4 J=I LR R8,R4 LR R9,R7 LOOPJ BXH R5,R8,ENDLOOPJ do J=I*2 to N by I LR R1,R5 R1=J BCTR R1,0 LA R14,CRIBLE(R1) MVI 0(R14),X'00' CRIBLE(J)='0'B B LOOPJ ENDLOOPJ EQU * ENDIF EQU * B LOOPI ENDLOOPI EQU * LA R4,1 I=1 LA R6,1 L R7,N LOOP BXH R4,R6,ENDLOOP do I=1 to N LR R1,R4 R1=I BCTR R1,0 LA R14,CRIBLE(R1) CLI 0(R14),X'01' BNE NOTPRIME if not CRIBLE(I) CVD R4,P P=I UNPK Z,P Z=P MVC C,Z C=Z OI C+L'C-1,X'F0' zap sign MVC WTOBUF(8),C+8 WTO MF=(E,WTOMSG) NOTPRIME EQU * B LOOP ENDLOOP EQU * RETURN EQU * LM R14,R12,12(R13) restore context XR R15,R15 set return code to 0 BR R14 return to caller
---- DATA
I DS F J DS F DS 0F P DS PL8 packed Z DS ZL16 zoned C DS CL16 character WTOMSG DS 0F DC H'80' length of WTO buffer DC H'0' must be binary zeroes WTOBUF DC 80C' ' LTORG N DC F'100000' CRIBLE DC 100000X'01' YREGS END ERATOST
<pre style="height:20ex">
00000002
00000003
00000005
00000007
00000011
00000013
00000017
00000019
00000023
00000029
00000031
00000037
00000041
00000043
00000047
00000053
00000059
00000061
00000067
...
00099767
00099787
00099793
00099809
00099817
00099823
00099829
00099833
00099839
00099859
00099871
00099877
00099881
00099901
00099907
00099923
00099929
00099961
00099971
00099989
00099991
6502 Assembly
If this subroutine is called with the value of n in the accumulator, it will store an array of the primes less than n beginning at address 1000 hex and return the number of primes it has found in the accumulator.
ERATOS: STA $D0 ; value of n
LDA #$00
LDX #$00
SETUP: STA $1000,X ; populate array
ADC #$01
INX
CPX $D0
BPL SET
JMP SETUP
SET: LDX #$02
SIEVE: LDA $1000,X ; find non-zero
INX
CPX $D0
BPL SIEVED
CMP #$00
BEQ SIEVE
STA $D1 ; current prime
MARK: CLC
ADC $D1
TAY
LDA #$00
STA $1000,Y
TYA
CMP $D0
BPL SIEVE
JMP MARK
SIEVED: LDX #$01
LDY #$00
COPY: INX
CPX $D0
BPL COPIED
LDA $1000,X
CMP #$00
BEQ COPY
STA $2000,Y
INY
JMP COPY
COPIED: TYA ; how many found
RTS
68000 Assembly
'''Algorithm somewhat optimized:''' array omits 1, 2, all higher odd numbers. Optimized for storage: uses bit array for prime/composite flags.
Some of the macro code is derived from the examples included with EASy68K. See 68000 "100 Doors" listing for additional information.
*-----------------------------------------------------------
* Title : BitSieve
* Written by : G. A. Tippery
* Date : 2014-Feb-24, 2013-Dec-22
* Description: Prime number sieve
*-----------------------------------------------------------
ORG $1000
** ---- Generic macros ---- **
PUSH MACRO
MOVE.L \1,-(SP)
ENDM
POP MACRO
MOVE.L (SP)+,\1
ENDM
DROP MACRO
ADDQ #4,SP
ENDM
PUTS MACRO
** Print a null-terminated string w/o CRLF **
** Usage: PUTS stringaddress
** Returns with D0, A1 modified
MOVEQ #14,D0 ; task number 14 (display null string)
LEA \1,A1 ; address of string
TRAP #15 ; display it
ENDM
GETN MACRO
MOVEQ #4,D0 ; Read a number from the keyboard into D1.L.
TRAP #15
ENDM
** ---- Application-specific macros ---- **
val MACRO ; Used by bit sieve. Converts bit address to the number it represents.
ADD.L \1,\1 ; double it because odd numbers are omitted
ADDQ #3,\1 ; add offset because initial primes (1, 2) are omitted
ENDM
* **
### ==========================================================================
**
* ** Integer square root routine, bisection method **
* ** IN: D0, should be 0<D0<$10000 (65536) -- higher values MAY work, no guarantee
* ** OUT: D1
*
SquareRoot:
*
MOVEM.L D2-D4,-(SP) ; save registers needed for local variables
* DO == n
* D1 == a
* D2 == b
* D3 == guess
* D4 == temp
*
* a = 1;
* b = n;
MOVEQ #1,D1
MOVE.L D0,D2
* do {
REPEAT
* guess = (a+b)/2;
MOVE.L D1,D3
ADD.L D2,D3
LSR.L #1,D3
* if (guess*guess > n) { // inverse function of sqrt is square
MOVE.L D3,D4
MULU D4,D4 ; guess^2
CMP.L D0,D4
BLS .else
* b = guess;
MOVE.L D3,D2
BRA .endif
* } else {
.else:
* a = guess;
MOVE.L D3,D1
* } //if
.endif:
* } while ((b-a) > 1); ; Same as until (b-a)<=1 or until (a-b)>=1
MOVE.L D2,D4
SUB.L D1,D4 ; b-a
UNTIL.L D4 <LE> #1 DO.S
* return (a) ; Result is in D1
* } //LongSqrt()
MOVEM.L (SP)+,D2-D4 ; restore saved registers
RTS
*
* **
### ==========================================================================
**
**
### =================================================================
**
*
** Prime-number Sieve of Eratosthenes routine using a big bit field for flags **
* Enter with D0 = size of sieve (bit array)
* Prints found primes 10 per line
* Returns # prime found in D6
*
* Register usage:
*
* D0 == n
* D1 == prime
* D2 == sqroot
* D3 == PIndex
* D4 == CIndex
* D5 == MaxIndex
* D6 == PCount
*
* A0 == PMtx[0]
*
* On return, all registers above except D0 are modified. Could add MOVEMs to save and restore D2-D6/A0.
*
** ------------------------ **
GetBit: ** sub-part of Sieve subroutine **
** Entry: bit # is on TOS
** Exit: A6 holds the byte number, D7 holds the bit number within the byte
** Note: Input param is still on TOS after return. Could have passed via a register, but
** wanted to practice with stack. :)
*
MOVE.L (4,SP),D7 ; get value from (pre-call) TOS
ASR.L #3,D7 ; /8
MOVEA D7,A6 ; byte #
MOVE.L (4,SP),D7 ; get value from (pre-call) TOS
AND.L #$7,D7 ; bit #
RTS
** ------------------------ **
Sieve:
MOVE D0,D5
SUBQ #1,D5
JSR SquareRoot ; sqrt D0 => D1
MOVE.L D1,D2
LEA PArray,A0
CLR.L D3
*
PrimeLoop:
MOVE.L D3,D1
val D1
MOVE.L D3,D4
ADD.L D1,D4
*
CxLoop: ; Goes through array marking multiples of d1 as composite numbers
CMP.L D5,D4
BHI ExitCx
PUSH D4 ; set D7 as bit # and A6 as byte pointer for D4'th bit of array
JSR GetBit
DROP
BSET D7,0(A0,A6.L) ; set bit to mark as composite number
ADD.L D1,D4 ; next number to mark
BRA CxLoop
ExitCx:
CLR.L D1 ; Clear new-prime-found flag
ADDQ #1,D3 ; Start just past last prime found
PxLoop: ; Searches for next unmarked (not composite) number
CMP.L D2,D3 ; no point searching past where first unmarked multiple would be past end of array
BHI ExitPx ; if past end of array
TST.L D1
BNE ExitPx ; if flag set, new prime found
PUSH D3 ; check D3'th bit flag
JSR GetBit ; sets D7 as bit # and A6 as byte pointer
DROP ; drop TOS
BTST D7,0(A0,A6.L) ; read bit flag
BNE IsSet ; If already tagged as composite
MOVEQ #-1,D1 ; Set flag that we've found a new prime
IsSet:
ADDQ #1,D3 ; next PIndex
BRA PxLoop
ExitPx:
SUBQ #1,D3 ; back up PIndex
TST.L D1 ; Did we find a new prime #?
BNE PrimeLoop ; If another prime # found, go process it
*
; fall through to print routine
** ------------------------ **
* Print primes found
*
* D4 == Column count
*
* Print header and assumed primes (#1, #2)
PUTS Header ; Print string @ Header, no CR/LF
MOVEQ #2,D6 ; Start counter at 2 because #1 and #2 are assumed primes
MOVEQ #2,D4
*
MOVEQ #0,D3
PrintLoop:
CMP.L D5,D3
BHS ExitPL
PUSH D3
JSR GetBit ; sets D7 as bit # and A6 as byte pointer
DROP ; drop TOS
BTST D7,0(A0,A6.L)
BNE NotPrime
* printf(" %6d", val(PIndex)
MOVE.L D3,D1
val D1
AND.L #$0000FFFF,D1
MOVEQ #6,D2
MOVEQ #20,D0 ; display signed RJ
TRAP #15
ADDQ #1,D4
ADDQ #1,D6
* *** Display formatting ***
* if((PCount % 10) == 0) printf("\n");
CMP #10,D4
BLO NoLF
PUTS CRLF
MOVEQ #0,D4
NoLF:
NotPrime:
ADDQ #1,D3
BRA PrintLoop
ExitPL:
RTS
**
### =================================================================
**
N EQU 5000 ; *** Size of boolean (bit) array ***
SizeInBytes EQU (N+7)/8
*
START: ; first instruction of program
MOVE.L #N,D0 ; # to test
JSR Sieve
* printf("\n %d prime numbers found.\n", D6); ***
PUTS Summary1,A1
MOVE #3,D0 ; Display signed number in D1.L in decimal in smallest field.
MOVE.W D6,D1
TRAP #15
PUTS Summary2,A1
SIMHALT ; halt simulator
**
### =================================================================
**
* Variables and constants here
ORG $2000
CR EQU 13
LF EQU 10
CRLF DC.B CR,LF,$00
PArray: DCB.B SizeInBytes,0
Header: DC.B CR,LF,LF,' Primes',CR,LF,' ======',CR,LF
DC.B ' 1 2',$00
Summary1: DC.B CR,LF,' ',$00
Summary2: DC.B ' prime numbers found.',CR,LF,$00
END START ; last line of source
ABAP
PARAMETERS: p_limit TYPE i OBLIGATORY DEFAULT 100.
AT SELECTION-SCREEN ON p_limit.
IF p_limit LE 1.
MESSAGE 'Limit must be higher then 1.' TYPE 'E'.
ENDIF.
START-OF-SELECTION.
FIELD-SYMBOLS: <fs_prime> TYPE flag.
DATA: gt_prime TYPE TABLE OF flag,
gv_prime TYPE flag,
gv_i TYPE i,
gv_j TYPE i.
DO p_limit TIMES.
IF sy-index > 1.
gv_prime = abap_true.
ELSE.
gv_prime = abap_false.
ENDIF.
APPEND gv_prime TO gt_prime.
ENDDO.
gv_i = 2.
WHILE ( gv_i <= trunc( sqrt( p_limit ) ) ).
IF ( gt_prime[ gv_i ] EQ abap_true ).
gv_j = gv_i ** 2.
WHILE ( gv_j <= p_limit ).
gt_prime[ gv_j ] = abap_false.
gv_j = ( gv_i ** 2 ) + ( sy-index * gv_i ).
ENDWHILE.
ENDIF.
gv_i = gv_i + 1.
ENDWHILE.
LOOP AT gt_prime INTO gv_prime.
IF gv_prime = abap_true.
WRITE: / sy-tabix.
ENDIF.
ENDLOOP.
ACL2
(defun nats-to-from (n i)
(declare (xargs :measure (nfix (- n i))))
(if (zp (- n i))
nil
(cons i (nats-to-from n (+ i 1)))))
(defun remove-multiples-up-to-r (factor limit xs i)
(declare (xargs :measure (nfix (- limit i))))
(if (or (> i limit)
(zp (- limit i))
(zp factor))
xs
(remove-multiples-up-to-r
factor
limit
(remove i xs)
(+ i factor))))
(defun remove-multiples-up-to (factor limit xs)
(remove-multiples-up-to-r factor limit xs (* factor 2)))
(defun sieve-r (factor limit)
(declare (xargs :measure (nfix (- limit factor))))
(if (zp (- limit factor))
(nats-to-from limit 2)
(remove-multiples-up-to factor (+ limit 1)
(sieve-r (1+ factor) limit))))
(defun sieve (limit)
(sieve-r 2 limit))
ActionScript
Works with ActionScript 3.0 (this is utilizing the actions panel, not a separated class file)
function eratosthenes(limit:int):Array
{
var primes:Array = new Array();
if (limit >= 2) {
var sqrtlmt:int = int(Math.sqrt(limit) - 2);
var nums:Array = new Array(); // start with an empty Array...
for (var i:int = 2; i <= limit; i++) // and
nums.push(i); // only initialize the Array once...
for (var j:int = 0; j <= sqrtlmt; j++) {
var p:int = nums[j]
if (p)
for (var t:int = p * p - 2; t < nums.length; t += p)
nums[t] = 0;
}
for (var m:int = 0; m < nums.length; m++) {
var r:int = nums[m];
if (r)
primes.push(r);
}
}
return primes;
}
var e:Array = eratosthenes(1000);
trace(e);
Output:
2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,293,307,311,313,317,331,337,347,349,353,359,367,373,379,383,389,397,401,409,419,421,431,433,439,443,449,457,461,463,467,479,487,491,499,503,509,521,523,541,547,557,563,569,571,577,587,593,599,601,607,613,617,619,631,641,643,647,653,659,661,673,677,683,691,701,709,719,727,733,739,743,751,757,761,769,773,787,797,809,811,821,823,827,829,839,853,857,859,863,877,881,883,887,907,911,919,929,937,941,947,953,967,971,977,983,991,997
Ada
with Ada.Text_IO, Ada.Command_Line;
procedure Eratos is
Last: Positive := Positive'Value(Ada.Command_Line.Argument(1));
Prime: array(1 .. Last) of Boolean := (1 => False, others => True);
Base: Positive := 2;
Cnt: Positive;
begin
loop
exit when Base * Base > Last;
if Prime(Base) then
Cnt := Base + Base;
loop
exit when Cnt > Last;
Prime(Cnt) := False;
Cnt := Cnt + Base;
end loop;
end if;
Base := Base + 1;
end loop;
Ada.Text_IO.Put("Primes less or equal" & Positive'Image(Last) &" are:");
for Number in Prime'Range loop
if Prime(Number) then
Ada.Text_IO.Put(Positive'Image(Number));
end if;
end loop;
end Eratos;
> ./eratos 31
Primes less or equal 31 are : 2 3 5 7 11 13 17 19 23 29 31
Agda
-- imports
open import Data.Nat as ℕ using (ℕ; suc; zero; _+_; _∸_)
open import Data.Vec as Vec using (Vec; _∷_; []; tabulate; foldr)
open import Data.Fin as Fin using (Fin; suc; zero)
open import Function using (_∘_; const; id)
open import Data.List as List using (List; _∷_; [])
open import Data.Maybe using (Maybe; just; nothing)
-- Without square cutoff optimization
module Simple where
primes : ∀ n → List (Fin n)
primes zero = []
primes (suc zero) = []
primes (suc (suc zero)) = []
primes (suc (suc (suc m))) = sieve (tabulate (just ∘ suc))
where
sieve : ∀ {n} → Vec (Maybe (Fin (2 + m))) n → List (Fin (3 + m))
sieve [] = []
sieve (nothing ∷ xs) = sieve xs
sieve (just x ∷ xs) = suc x ∷ sieve (foldr B remove (const []) xs x)
where
B = λ n → ∀ {i} → Fin i → Vec (Maybe (Fin (2 + m))) n
remove : ∀ {n} → Maybe (Fin (2 + m)) → B n → B (suc n)
remove _ ys zero = nothing ∷ ys x
remove y ys (suc z) = y ∷ ys z
-- With square cutoff optimization
module SquareOpt where
primes : ∀ n → List (Fin n)
primes zero = []
primes (suc zero) = []
primes (suc (suc zero)) = []
primes (suc (suc (suc m))) = sieve 1 m (Vec.tabulate (just ∘ Fin.suc ∘ Fin.suc))
where
sieve : ∀ {n} → ℕ → ℕ → Vec (Maybe (Fin (3 + m))) n → List (Fin (3 + m))
sieve _ zero = List.mapMaybe id ∘ Vec.toList
sieve _ (suc _) [] = []
sieve i (suc l) (nothing ∷ xs) = sieve (suc i) (l ∸ i ∸ i) xs
sieve i (suc l) (just x ∷ xs) = x ∷ sieve (suc i) (l ∸ i ∸ i) (Vec.foldr B remove (const []) xs i)
where
B = λ n → ℕ → Vec (Maybe (Fin (3 + m))) n
remove : ∀ {i} → Maybe (Fin (3 + m)) → B i → B (suc i)
remove _ ys zero = nothing ∷ ys i
remove y ys (suc j) = y ∷ ys j
Agena
Tested with Agena 2.9.5 Win32
# Sieve of Eratosthenes
# generate and return a sequence containing the primes up to sieveSize
sieve := proc( sieveSize :: number ) :: sequence is
local sieve, result;
result := seq(); # sequence of primes - initially empty
create register sieve( sieveSize ); # "vector" to be sieved
sieve[ 1 ] := false;
for sPos from 2 to sieveSize do sieve[ sPos ] := true od;
# sieve the primes
for sPos from 2 to entier( sqrt( sieveSize ) ) do
if sieve[ sPos ] then
for p from sPos * sPos to sieveSize by sPos do
sieve[ p ] := false
od
fi
od;
# construct the sequence of primes
for sPos from 1 to sieveSize do
if sieve[ sPos ] then insert sPos into result fi
od
return result
end; # sieve
# test the sieve proc
for i in sieve( 100 ) do write( " ", i ) od; print();
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
ALGOL 60
'''Based on the 1962 Revised Repport''': '''comment''' Sieve of Eratosthenes; '''begin''' '''integer array''' t[0:1000]; '''integer''' i,j,k; '''for''' i:=0 '''step''' 1 '''until''' 1000 '''do''' t[i]:=1; t[0]:=0; t[1]:=0; i:=0; '''for''' i:=i '''while''' i<1000 '''do''' '''begin''' '''for''' i:=i '''while''' i<1000 '''and''' t[i]=0 '''do''' i:=i+1; '''if''' i<1000 '''then''' '''begin''' j:=2; k:=ji; '''for''' k:=k '''while''' k<1000 '''do''' '''begin''' t[k]:=0; j:=j+1; k:=ji '''end'''; i:=i+1 '''end''' '''end'''; '''for''' i:=0 '''step''' 1 '''until''' 999 '''do''' '''if''' t[i]≠0 '''then''' print(i,ꞌ is primeꞌ) '''end'''
'''An 1964 Implementation''':
'''Works with:''' ALGOL 60 for OS/360
'BEGIN'
'INTEGER' 'ARRAY' CANDIDATES(/0..1000/);
'INTEGER' I,J,K;
'COMMENT' SET LINE-LENGTH=120,SET LINES-PER-PAGE=62,OPEN;
SYSACT(1,6,120); SYSACT(1,8,62); SYSACT(1,12,1);
'FOR' I := 0 'STEP' 1 'UNTIL' 1000 'DO'
'BEGIN'
CANDIDATES(/I/) := 1;
'END';
CANDIDATES(/0/) := 0;
CANDIDATES(/1/) := 0;
I := 0;
'FOR' I := I 'WHILE' I 'LESS' 1000 'DO'
'BEGIN'
'FOR' I := I 'WHILE' I 'LESS' 1000
'AND' CANDIDATES(/I/) 'EQUAL' 0 'DO'
I := I+1;
'IF' I 'LESS' 1000 'THEN'
'BEGIN'
J := 2;
K := J*I;
'FOR' K := K 'WHILE' K 'LESS' 1000 'DO'
'BEGIN'
CANDIDATES(/K/) := 0;
J := J + 1;
K := J*I;
'END';
I := I+1;
'END'
'END';
'FOR' I := 0 'STEP' 1 'UNTIL' 999 'DO'
'IF' CANDIDATES(/I/) 'NOTEQUAL' 0 'THEN'
'BEGIN'
OUTINTEGER(1,I);
OUTSTRING(1,'(' IS PRIME')');
'COMMENT' NEW LINE;
SYSACT(1,14,1)
'END'
'END'
'END'
ALGOL 68
BOOL prime = TRUE, non prime = FALSE;
PROC eratosthenes = (INT n)[]BOOL:
(
[n]BOOL sieve;
FOR i TO UPB sieve DO sieve[i] := prime OD;
INT m = ENTIER sqrt(n);
sieve[1] := non prime;
FOR i FROM 2 TO m DO
IF sieve[i] = prime THEN
FOR j FROM i*i BY i TO n DO
sieve[j] := non prime
OD
FI
OD;
sieve
);
print((eratosthenes(80),new line))
FTTFTFTFFFTFTFFFTFTFFFTFFFFFTFTFFFFFTFFFTFTFFFTFFFFFTFFFFFTFTFFFFFTFFFTFTFFFFFTF
=={{header|ALGOL-M}}==
BEGIN
COMMENT
FIND PRIMES UP TO THE SPECIFIED LIMIT (HERE 1,000) USING
THE SIEVE OF ERATOSTHENES;
% CALCULATE INTEGER SQUARE ROOT %
INTEGER FUNCTION ISQRT(N);
INTEGER N;
BEGIN
INTEGER R1, R2;
R1 := N;
R2 := 1;
WHILE R1 > R2 DO
BEGIN
R1 := (R1+R2) / 2;
R2 := N / R1;
END;
ISQRT := R1;
END;
INTEGER LIMIT, I, J, FALSE, TRUE, COL, COUNT;
INTEGER ARRAY FLAGS[1:1000];
LIMIT := 1000;
FALSE := 0;
TRUE := 1;
WRITE("FINDING PRIMES FROM 2 TO",LIMIT);
% INITIALIZE TABLE %
FOR I := 1 STEP 1 UNTIL LIMIT DO
FLAGS[I] := TRUE;
% SIEVE FOR PRIMES %
FOR I := 2 STEP 1 UNTIL ISQRT(LIMIT) DO
BEGIN
IF FLAGS[I] = TRUE THEN
FOR J := (I * I) STEP I UNTIL LIMIT DO
FLAGS[J] := FALSE;
END;
% WRITE OUT THE PRIMES EIGHT PER LINE %
COUNT := 0;
COL := 1;
WRITE(" ");
FOR I := 2 STEP 1 UNTIL LIMIT DO
BEGIN
IF FLAGS[I] = TRUE THEN
BEGIN
WRITEON(I);
COUNT := COUNT + 1;
COL := COL + 1;
IF COL > 8 THEN
BEGIN
WRITE(" ");
COL := 1;
END;
END;
END;
WRITE(" ");
WRITE(COUNT, " primes were found.");
END
ALGOL W
begin
% implements the sieve of Eratosthenes %
% s(i) is set to true if i is prime, false otherwise %
% algol W doesn't have a upb operator, so we pass the size of the %
% array in n %
procedure sieve( logical array s ( * ); integer value n ) ;
begin
% start with everything flagged as prime %
for i := 1 until n do s( i ) := true;
% sieve out the non-primes %
s( 1 ) := false;
for i := 2 until truncate( sqrt( n ) )
do begin
if s( i )
then begin
for p := i * i step i until n do s( p ) := false
end if_s_i
end for_i ;
end sieve ;
% test the sieve procedure %
integer sieveMax;
sieveMax := 100;
begin
logical array s ( 1 :: sieveMax );
i_w := 2; % set output field width %
s_w := 1; % and output separator width %
% find and display the primes %
sieve( s, sieveMax );
for i := 1 until sieveMax do if s( i ) then writeon( i );
end
end.
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
=={{Header|APL}}==
All these versions requires ⎕io←0 (index origin 0).
It would have been better to require a result of the boolean mask rather than the actual list of primes. The list of primes obtains readily from the mask by application of a simple function (here {⍵/⍳⍴⍵}). Other related computations (such as the number of primes < n) obtain readily from the mask, easier than producing the list of primes.
=== Non-Optimized Version ===
sieve2←{
b←⍵⍴1
b[⍳2⌊⍵]←0
2≥⍵:b
p←{⍵/⍳⍴⍵}∇⌈⍵*0.5
m←1+⌊(⍵-1+p×p)÷p
b ⊣ p {b[⍺×⍺+⍳⍵]←0}¨ m
}
primes2←{⍵/⍳⍴⍵}∘sieve2
The required list of prime divisors obtains by recursion ({⍵/⍳⍴⍵}∇⌈⍵*0.5).
Optimized Version
sieve←{
b←⍵⍴{∧⌿↑(×/⍵)⍴¨~⍵↑¨1}2 3 5
b[⍳6⌊⍵]←(6⌊⍵)⍴0 0 1 1 0 1
49≥⍵:b
p←3↓{⍵/⍳⍴⍵}∇⌈⍵*0.5
m←1+⌊(⍵-1+p×p)÷2×p
b ⊣ p {b[⍺×⍺+2×⍳⍵]←0}¨ m
}
primes←{⍵/⍳⍴⍵}∘sieve
The optimizations are as follows:
- Multiples of 2 3 5 are marked by initializing b with ⍵⍴{∧⌿↑(×/⍵)⍴¨~⍵↑¨1}2 3 5 rather than with ⍵⍴1.
- Subsequently, only odd multiples of primes > 5 are marked.
- Multiples of a prime to be marked start at its square.
Examples
primes 100
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
primes¨ ⍳14
┌┬┬┬─┬───┬───┬─────┬─────┬───────┬───────┬───────┬───────┬──────────┬──────────┐
││││2│2 3│2 3│2 3 5│2 3 5│2 3 5 7│2 3 5 7│2 3 5 7│2 3 5 7│2 3 5 7 11│2 3 5 7 11│
└┴┴┴─┴───┴───┴─────┴─────┴───────┴───────┴───────┴───────┴──────────┴──────────┘
sieve 13
0 0 1 1 0 1 0 1 0 0 0 1 0
+/∘sieve¨ 10*⍳10
0 4 25 168 1229 9592 78498 664579 5761455 50847534
The last expression computes the number of primes < 1e0 1e1 ... 1e9. The last number 50847534 can perhaps be called the anti-Bertelsen's number (http://mathworld.wolfram.com/BertelsensNumber.html).
=={{Header|AppleScript}}== Note: This version of Trial Division has something like O(n^(3/2) asymptotic execution complexity rather than O(n log (log n)) for the true sieve.
to sieve(N)
script
on array()
set L to {}
repeat with i from 1 to N
set end of L to i
end repeat
L
end array
end script
set L to result's array()
set item 1 of L to false
repeat with x in L
repeat with y in L
try
if (x < y ^ 2) then exit repeat
if (x mod y = 0) then
set x's contents to false
exit repeat
end if
end try
end repeat
end repeat
numbers in L
end sieve
sieve(1000)
{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997}
=={{Header|AutoHotkey}}== {{AutoHotkey case}}Source: [http://www.autohotkey.com/forum/topic44657.html AutoHotkey forum] by Laszlo
MsgBox % "12345678901234567890`n" Sieve(20)
Sieve(n) { ; Sieve of Eratosthenes => string of 0|1 chars, 1 at position k: k is prime
Static zero := 48, one := 49 ; Asc("0"), Asc("1")
VarSetCapacity(S,n,one)
NumPut(zero,S,0,"char")
i := 2
Loop % sqrt(n)-1 {
If (NumGet(S,i-1,"char") = one)
Loop % n//i
If (A_Index > 1)
NumPut(zero,S,A_Index*i-1,"char")
i += 1+(i>2)
}
Return S
}
=={{Header|AutoIt}}==
$M = InputBox("Integer", "Enter biggest Integer")
Global $a[$M], $r[$M], $c = 1
For $i = 2 To $M -1
If Not $a[$i] Then
$r[$c] = $i
$c += 1
For $k = $i To $M -1 Step $i
$a[$k] = True
Next
EndIf
Next
$r[0] = $c - 1
ReDim $r[$c]
_ArrayDisplay($r)
AWK
An initial array holds all numbers 2..max (which is entered on stdin); then all products of integers are deleted from it; the remaining are displayed in the unsorted appearance of a hash table. Here, the script is entered directly on the commandline, and input entered on stdin: $ awk '{for(i=2;i<=$1;i++) a[i]=1;
for(i=2;i<=sqrt($1);i++) for(j=2;j<=$1;j++) delete a[i*j]; for(i in a) printf i" "}'
100 71 53 17 5 73 37 19 83 47 29 7 67 59 11 97 79 89 31 13 41 23 2 61 43 3
The following variant does not unset non-primes, but sets them to 0, to preserve order in output: $ awk '{for(i=2;i<=$1;i++) a[i]=1;
for(i=2;i<=sqrt($1);i++) for(j=2;j<=$1;j++) a[i*j]=0; for(i=2;i<=$1;i++) if(a[i])printf i" "}'
100 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
Now with the script from a file, input from commandline as well as stdin, and input is checked for valid numbers:
# usage: gawk -v n=101 -f sieve.awk
function sieve(n) { # print n,":"
for(i=2; i<=n; i++) a[i]=1;
for(i=2; i<=sqrt(n);i++) for(j=2;j<=n;j++) a[i*j]=0;
for(i=2; i<=n; i++) if(a[i]) printf i" "
print ""
}
BEGIN { print "Sieve of Eratosthenes:"
if(n>1) sieve(n)
}
{ n=$1+0 }
n<2 { exit }
{ sieve(n) }
END { print "Bye!" }
Here is an alternate version that uses an associative array to record composites with a prime dividing it. It can be considered a slow version, as it does not cross out composites until needed. This version assumes enough memory to hold all primes up to ULIMIT. It prints out noncomposites greater than 1.
BEGIN { ULIMIT=100
for ( n=1 ; (n++) < ULIMIT ; )
if (n in S) {
p = S[n]
delete S[n]
for ( m = n ; (m += p) in S ; ) { }
S[m] = p
}
else print ( S[(n+n)] = n )
}
==Bash== ''See solutions at [[{{FULLPAGENAME}}#UNIX Shell|UNIX Shell]].''
BASIC
DIM n AS Integer, k AS Integer, limit AS Integer
INPUT "Enter number to search to: "; limit
DIM flags(limit) AS Integer
FOR n = 2 TO SQR(limit)
IF flags(n) = 0 THEN
FOR k = n*n TO limit STEP n
flags(k) = 1
NEXT k
END IF
NEXT n
' Display the primes
FOR n = 2 TO limit
IF flags(n) = 0 THEN PRINT n; ", ";
NEXT n
=
Applesoft BASIC
=
10 INPUT "ENTER NUMBER TO SEARCH TO: ";LIMIT
20 DIM FLAGS(LIMIT)
30 FOR N = 2 TO SQR (LIMIT)
40 IF FLAGS(N) < > 0 GOTO 80
50 FOR K = N * N TO LIMIT STEP N
60 FLAGS(K) = 1
70 NEXT K
80 NEXT N
90 REM DISPLAY THE PRIMES
100 FOR N = 2 TO LIMIT
110 IF FLAGS(N) = 0 THEN PRINT N;", ";
120 NEXT N
==={{header|IS-BASIC}}===
=
## Locomotive Basic
=
```locobasic
10 DEFINT a-z
20 INPUT "Limit";limit
30 DIM f(limit)
40 FOR n=2 TO SQR(limit)
50 IF f(n)=1 THEN 90
60 FOR k=n*n TO limit STEP n
70 f(k)=1
80 NEXT k
90 NEXT n
100 FOR n=2 TO limit
110 IF f(n)=0 THEN PRINT n;",";
120 NEXT
=
MSX Basic
=
5 Rem MSX BRRJPA
10 INPUT "Search until: ";L
20 DIM p(L)
30 FOR n=2 TO SQR (L+1000)
40 IF p(n)<>0 THEN goto 80
50 FOR k=n*n TO L STEP n
60 LET p(k)=1
70 NEXT k
80 NEXT n
90 FOR n=2 TO L
100 IF p(n)=0 THEN PRINT n;", ";
110 NEXT n
=
Sinclair ZX81 BASIC
= If you only have 1k of RAM, this program will work—but you will only be able to sieve numbers up to 101. The program is therefore more useful if you have more memory available.
A note on FAST and SLOW: under normal circumstances the CPU spends about 3/4 of its time driving the display and only 1/4 doing everything else. Entering FAST mode blanks the screen (which we do not want to update anyway), resulting in substantially improved performance; we then return to SLOW mode when we have something to print out.
10 INPUT L
20 FAST
30 DIM N(L)
40 FOR I=2 TO SQR L
50 IF N(I) THEN GOTO 90
60 FOR J=I+I TO L STEP I
70 LET N(J)=1
80 NEXT J
90 NEXT I
100 SLOW
110 FOR I=2 TO L
120 IF NOT N(I) THEN PRINT I;" ";
130 NEXT I
=
ZX Spectrum Basic
=
10 INPUT "Enter number to search to: ";l
20 DIM p(l)
30 FOR n=2 TO SQR l
40 IF p(n)<>0 THEN NEXT n
50 FOR k=n*n TO l STEP n
60 LET p(k)=1
70 NEXT k
80 NEXT n
90 REM Display the primes
100 FOR n=2 TO l
110 IF p(n)=0 THEN PRINT n;", ";
120 NEXT n
BBC BASIC
limit% = 100000
DIM sieve% limit%
prime% = 2
WHILE prime%^2 < limit%
FOR I% = prime%*2 TO limit% STEP prime%
sieve%?I% = 1
NEXT
REPEAT prime% += 1 : UNTIL sieve%?prime%=0
ENDWHILE
REM Display the primes:
FOR I% = 1 TO limit%
IF sieve%?I% = 0 PRINT I%;
NEXT
Batch File
:: Sieve of Eratosthenes for Rosetta Code - PG
@echo off
setlocal ENABLEDELAYEDEXPANSION
setlocal ENABLEEXTENSIONS
rem echo on
set /p n=limit:
rem set n=100
for /L %%i in (1,1,%n%) do set crible.%%i=1
for /L %%i in (2,1,%n%) do (
if !crible.%%i! EQU 1 (
set /A w = %%i * 2
for /L %%j in (!w!,%%i,%n%) do (
set crible.%%j=0
)
)
)
for /L %%i in (2,1,%n%) do (
if !crible.%%i! EQU 1 echo %%i
)
pause
limit: 100
2
3
5
7
11
13
17
19
23
29
31
37
41
43
47
53
59
61
67
71
73
79
83
89
97
```
## Befunge
2>:3g" "-!v\ g30 <
|!`"O":+1_:.:03p>03g+:"O"`|
@ ^ p3\" ":<
2 234567890123456789012345678901234567890123456789012345678901234567890123456789
## Bracmat
This solution does not use an array. Instead, numbers themselves are used as variables. The numbers that are not prime are set (to the silly value "nonprime"). Finally all numbers up to the limit are tested for being initialised. The uninitialised (unset) ones must be the primes.
```bracmat
( ( eratosthenes
= n j i
. !arg:?n
& 1:?i
& whl
' ( (1+!i:?i)^2:~>!n:?j
& ( !!i
| whl
' ( !j:~>!n
& nonprime:?!j
& !j+!i:?j
)
)
)
& 1:?i
& whl
' ( 1+!i:~>!n:?i
& (!!i|put$(!i " "))
)
)
& eratosthenes$100
)
```
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
## C
Plain sieve, without any optimizations:
```cpp
#include
#include
char*
eratosthenes(int n, int *c)
{
char* sieve;
int i, j, m;
if(n < 2)
return NULL;
*c = n-1; /* primes count */
m = (int) sqrt((double) n);
/* calloc initializes to zero */
sieve = calloc(n+1,sizeof(char));
sieve[0] = 1;
sieve[1] = 1;
for(i = 2; i <= m; i++)
if(!sieve[i])
for (j = i*i; j <= n; j += i)
if(!sieve[j]){
sieve[j] = 1;
--(*c);
}
return sieve;
}
```
Possible optimizations include sieving only odd numbers (or more complex wheels), packing the sieve into bits to improve locality (and allow larger sieves), etc.
'''Another example:'''
We first fill ones into an array and assume all numbers are prime.
Then, in a loop, fill zeroes into those places where i * j is less than or equal to n (number of primes requested), which means they have multiples!
To understand this better, look at the output of the following example.
To print this back, we look for ones in the array and only print those spots.
```c
#include
#include
void sieve(int *, int);
int main(int argc, char *argv)
{
int *array, n=10;
array =(int *)malloc((n + 1) * sizeof(int));
sieve(array,n);
return 0;
}
void sieve(int *a, int n)
{
int i=0, j=0;
for(i=2; i<=n; i++) {
a[i] = 1;
}
for(i=2; i<=n; i++) {
printf("\ni:%d", i);
if(a[i] == 1) {
for(j=i; (i*j)<=n; j++) {
printf ("\nj:%d", j);
printf("\nBefore a[%d*%d]: %d", i, j, a[i*j]);
a[(i*j)] = 0;
printf("\nAfter a[%d*%d]: %d", i, j, a[i*j]);
}
}
}
printf("\nPrimes numbers from 1 to %d are : ", n);
for(i=2; i<=n; i++) {
if(a[i] == 1)
printf("%d, ", i);
}
printf("\n\n");
}
```
```Shell
i:2
j:2
Before a[2*2]: 1
After a[2*2]: 0
j:3
Before a[2*3]: 1
After a[2*3]: 0
j:4
Before a[2*4]: 1
After a[2*4]: 0
j:5
Before a[2*5]: 1
After a[2*5]: 0
i:3
j:3
Before a[3*3]: 1
After a[3*3]: 0
i:4
i:5
i:6
i:7
i:8
i:9
i:10
Primes numbers from 1 to 10 are : 2, 3, 5, 7,
```
## C++
### Standard Library
This implementation follows the standard library pattern of [http://en.cppreference.com/w/cpp/algorithm/iota std::iota]. The start and end iterators are provided for the container. The destination container is used for marking primes and then filled with the primes which are less than the container size. This method requires no memory allocation inside the function.
```cpp
#include
#include
#include
// requires Iterator satisfies RandomAccessIterator
template
size_t prime_sieve(Iterator start, Iterator end)
{
if (start == end) return 0;
// clear the container with 0
std::fill(start, end, 0);
// mark composites with 1
for (Iterator prime_it = start + 1; prime_it != end; ++prime_it)
{
if (*prime_it == 1) continue;
// determine the prime number represented by this iterator location
size_t stride = (prime_it - start) + 1;
// mark all multiples of this prime number up to max
Iterator mark_it = prime_it;
while ((end - mark_it) > stride)
{
std::advance(mark_it, stride);
*mark_it = 1;
}
}
// copy marked primes into container
Iterator out_it = start;
for (Iterator it = start + 1; it != end; ++it)
{
if (*it == 0)
{
*out_it = (it - start) + 1;
++out_it;
}
}
return out_it - start;
}
int main(int argc, const char* argv[])
{
std::vector primes(100);
size_t count = prime_sieve(primes.begin(), primes.end());
// display the primes
for (size_t i = 0; i < count; ++i)
std::cout << primes[i] << " ";
std::cout << std::endl;
return 1;
}
```
### Boost
```cpp
// yield all prime numbers less than limit.
template
void primesupto(int limit, UnaryFunction yield)
{
std::vector is_prime(limit, true);
const int sqrt_limit = static_cast(std::sqrt(limit));
for (int n = 2; n <= sqrt_limit; ++n)
if (is_prime[n]) {
yield(n);
for (unsigned k = n*n, ulim = static_cast(limit); k < ulim; k += n)
//NOTE: "unsigned" is used to avoid an overflow in `k+=n` for `limit` near INT_MAX
is_prime[k] = false;
}
for (int n = sqrt_limit + 1; n < limit; ++n)
if (is_prime[n])
yield(n);
}
```
Full program:
```cpp
/**
$ g++ -I/path/to/boost sieve.cpp -o sieve && sieve 10000000
*/
#include // uintmax_t
#include
#include
#include
#include
#include
#include
int main(int argc, char *argv[])
{
using namespace std;
using namespace boost::lambda;
int limit = 10000;
if (argc == 2) {
stringstream ss(argv[--argc]);
ss >> limit;
if (limit < 1 or ss.fail()) {
cerr << "USAGE:\n sieve LIMIT\n\nwhere LIMIT in the range [1, "
<< numeric_limits::max() << ")" << endl;
return 2;
}
}
// print primes less then 100
primesupto(100, cout << _1 << " ");
cout << endl;
// find number of primes less then limit and their sum
int count = 0;
uintmax_t sum = 0;
primesupto(limit, (var(sum) += _1, var(count) += 1));
cout << "limit sum pi(n)\n"
<< limit << " " << sum << " " << count << endl;
}
```
## C#
```c#
using System;
using System.Collections;
using System.Collections.Generic;
namespace SieveOfEratosthenes
{
class Program
{
static void Main(string[] args)
{
int maxprime = int.Parse(args[0]);
var primelist = GetAllPrimesLessThan(maxprime);
foreach (int prime in primelist)
{
Console.WriteLine(prime);
}
Console.WriteLine("Count = " + primelist.Count);
Console.ReadLine();
}
private static List GetAllPrimesLessThan(int maxPrime)
{
var primes = new List();
var maxSquareRoot = (int)Math.Sqrt(maxPrime);
var eliminated = new BitArray(maxPrime + 1);
for (int i = 2; i <= maxPrime; ++i)
{
if (!eliminated[i])
{
primes.Add(i);
if (i <= maxSquareRoot)
{
for (int j = i * i; j <= maxPrime; j += i)
{
eliminated[j] = true;
}
}
}
}
return primes;
}
}
}
```
### Unbounded
'''Richard Bird Sieve'''
To show that C# code can be written in somewhat functional paradigms, the following in an implementation of the Richard Bird sieve from the Epilogue of [Melissa E. O'Neill's definitive article](http://www.cs.hmc.edu/~oneill/papers/Sieve-JFP.pdf) in Haskell:
```c#
using System;
using System.Collections;
using System.Collections.Generic;
using System.Linq;
using PrimeT = System.UInt32;
class PrimesBird : IEnumerable {
private struct CIS {
public T v; public Func> cont;
public CIS(T v, Func> cont) {
this.v = v; this.cont = cont;
}
}
private CIS pmlts(PrimeT p) {
Func> fn = null;
fn = (c) => new CIS(c, () => fn(c + p));
return fn(p * p);
}
private CIS> allmlts(CIS ps) {
return new CIS>(pmlts(ps.v), () => allmlts(ps.cont())); }
private CIS merge(CIS xs, CIS ys) {
var x = xs.v; var y = ys.v;
if (x < y) return new CIS(x, () => merge(xs.cont(), ys));
else if (y < x) return new CIS(y, () => merge(xs, ys.cont()));
else return new CIS(x, () => merge(xs.cont(), ys.cont()));
}
private CIS cmpsts(CIS> css) {
return new CIS(css.v.v, () => merge(css.v.cont(), cmpsts(css.cont()))); }
private CIS minusat(PrimeT n, CIS cs) {
var nn = n; var ncs = cs;
for (; ; ++nn) {
if (nn >= ncs.v) ncs = ncs.cont();
else return new CIS(nn, () => minusat(++nn, ncs));
}
}
private CIS prms() {
return new CIS(2, () => minusat(3, cmpsts(allmlts(prms())))); }
public IEnumerator GetEnumerator() {
for (var ps = prms(); ; ps = ps.cont()) yield return ps.v;
}
IEnumerator IEnumerable.GetEnumerator() { return (IEnumerator)GetEnumerator(); }
}
```
'''Tree Folding Sieve'''
The above code can easily be converted to "'''odds-only'''" and a infinite tree-like folding scheme with the following minor changes:
```c#
using System;
using System.Collections;
using System.Collections.Generic;
using System.Linq;
using PrimeT = System.UInt32;
class PrimesTreeFold : IEnumerable {
private struct CIS {
public T v; public Func> cont;
public CIS(T v, Func> cont) {
this.v = v; this.cont = cont;
}
}
private CIS pmlts(PrimeT p) {
var adv = p + p;
Func> fn = null;
fn = (c) => new CIS(c, () => fn(c + adv));
return fn(p * p);
}
private CIS> allmlts(CIS ps) {
return new CIS>(pmlts(ps.v), () => allmlts(ps.cont()));
}
private CIS merge(CIS xs, CIS ys) {
var x = xs.v; var y = ys.v;
if (x < y) return new CIS(x, () => merge(xs.cont(), ys));
else if (y < x) return new CIS(y, () => merge(xs, ys.cont()));
else return new CIS(x, () => merge(xs.cont(), ys.cont()));
}
private CIS> pairs(CIS> css) {
var nxtcss = css.cont();
return new CIS>(merge(css.v, nxtcss.v), () => pairs(nxtcss.cont())); }
private CIS cmpsts(CIS> css) {
return new CIS(css.v.v, () => merge(css.v.cont(), cmpsts(pairs(css.cont()))));
}
private CIS minusat(PrimeT n, CIS cs) {
var nn = n; var ncs = cs;
for (; ; nn += 2) {
if (nn >= ncs.v) ncs = ncs.cont();
else return new CIS(nn, () => minusat(nn + 2, ncs));
}
}
private CIS oddprms() {
return new CIS(3, () => minusat(5, cmpsts(allmlts(oddprms()))));
}
public IEnumerator GetEnumerator() {
yield return 2;
for (var ps = oddprms(); ; ps = ps.cont()) yield return ps.v;
}
IEnumerator IEnumerable.GetEnumerator() { return (IEnumerator)GetEnumerator(); }
}
```
The above code runs over ten times faster than the original Richard Bird algorithm.
'''Priority Queue Sieve'''
First, an implementation of a Min Heap Priority Queue is provided as extracted from the entry at [http://rosettacode.org/wiki/Priority_queue#C.23 RosettaCode], with only the necessary methods duplicated here:
```c#
namespace PriorityQ {
using KeyT = System.UInt32;
using System;
using System.Collections.Generic;
using System.Linq;
class Tuple { // for DotNet 3.5 without Tuple's
public K Item1; public V Item2;
public Tuple(K k, V v) { Item1 = k; Item2 = v; }
public override string ToString() {
return "(" + Item1.ToString() + ", " + Item2.ToString() + ")";
}
}
class MinHeapPQ {
private struct HeapEntry {
public KeyT k; public V v;
public HeapEntry(KeyT k, V v) { this.k = k; this.v = v; }
}
private List pq;
private MinHeapPQ() { this.pq = new List(); }
private bool mt { get { return pq.Count == 0; } }
private Tuple pkmn {
get {
if (pq.Count == 0) return null;
else {
var mn = pq[0];
return new Tuple(mn.k, mn.v);
}
}
}
private void psh(KeyT k, V v) { // add extra very high item if none
if (pq.Count == 0) pq.Add(new HeapEntry(UInt32.MaxValue, v));
var i = pq.Count; pq.Add(pq[i - 1]); // copy bottom item...
for (var ni = i >> 1; ni > 0; i >>= 1, ni >>= 1) {
var t = pq[ni - 1];
if (t.k > k) pq[i - 1] = t; else break;
}
pq[i - 1] = new HeapEntry(k, v);
}
private void siftdown(KeyT k, V v, int ndx) {
var cnt = pq.Count - 1; var i = ndx;
for (var ni = i + i + 1; ni < cnt; ni = ni + ni + 1) {
var oi = i; var lk = pq[ni].k; var rk = pq[ni + 1].k;
var nk = k;
if (k > lk) { i = ni; nk = lk; }
if (nk > rk) { ni += 1; i = ni; }
if (i != oi) pq[oi] = pq[i]; else break;
}
pq[i] = new HeapEntry(k, v);
}
private void rplcmin(KeyT k, V v) {
if (pq.Count > 1) siftdown(k, v, 0); }
public static MinHeapPQ empty { get { return new MinHeapPQ(); } }
public static Tuple peekMin(MinHeapPQ pq) { return pq.pkmn; }
public static MinHeapPQ push(KeyT k, V v, MinHeapPQ pq) {
pq.psh(k, v); return pq; }
public static MinHeapPQ replaceMin(KeyT k, V v, MinHeapPQ pq) {
pq.rplcmin(k, v); return pq; }
}
```
The following code implements an improved version of the '''odds-only''' O'Neil algorithm, which provides the improvements of only adding base prime composite number streams to the queue when the sieved number reaches the square of the base prime (saving a huge amount of memory and considerable execution time, including not needing as wide a range of a type for the internal prime numbers) as well as minimizing stream processing using fusion:
```c#
using System;
using System.Collections;
using System.Collections.Generic;
using System.Linq;
using PrimeT = System.UInt32;
class PrimesPQ : IEnumerable {
private IEnumerator nmrtr() {
MinHeapPQ pq = MinHeapPQ.empty;
PrimeT bp = 3; PrimeT q = 9;
IEnumerator bps = null;
yield return 2; yield return 3;
for (var n = (PrimeT)5; ; n += 2) {
if (n >= q) { // always equal or less...
if (q <= 9) {
bps = nmrtr();
bps.MoveNext(); bps.MoveNext(); } // move to 3...
bps.MoveNext(); var nbp = bps.Current; q = nbp * nbp;
var adv = bp + bp; bp = nbp;
pq = MinHeapPQ.push(n + adv, adv, pq);
}
else {
var pk = MinHeapPQ.peekMin(pq);
var ck = (pk == null) ? q : pk.Item1;
if (n >= ck) {
do { var adv = pk.Item2;
pq = MinHeapPQ.replaceMin(ck + adv, adv, pq);
pk = MinHeapPQ.peekMin(pq); ck = pk.Item1;
} while (n >= ck);
}
else yield return n;
}
}
}
public IEnumerator GetEnumerator() { return nmrtr(); }
IEnumerator IEnumerable.GetEnumerator() { return (IEnumerator)GetEnumerator(); }
}
```
The above code is at least about 2.5 times faster than the Tree Folding version.
'''Dictionary (Hash table) Sieve'''
The above code adds quite a bit of overhead in having to provide a version of a Priority Queue for little advantage over a Dictionary (hash table based) version as per the code below:
```c#
using System;
using System.Collections;
using System.Collections.Generic;
using System.Linq;
using PrimeT = System.UInt32;
class PrimesDict : IEnumerable {
private IEnumerator nmrtr() {
Dictionary dct = new Dictionary();
PrimeT bp = 3; PrimeT q = 9;
IEnumerator bps = null;
yield return 2; yield return 3;
for (var n = (PrimeT)5; ; n += 2) {
if (n >= q) { // always equal or less...
if (q <= 9) {
bps = nmrtr();
bps.MoveNext(); bps.MoveNext();
} // move to 3...
bps.MoveNext(); var nbp = bps.Current; q = nbp * nbp;
var adv = bp + bp; bp = nbp;
dct.Add(n + adv, adv);
}
else {
if (dct.ContainsKey(n)) {
PrimeT nadv; dct.TryGetValue(n, out nadv); dct.Remove(n); var nc = n + nadv;
while (dct.ContainsKey(nc)) nc += nadv;
dct.Add(nc, nadv);
}
else yield return n;
}
}
}
public IEnumerator GetEnumerator() { return nmrtr(); }
IEnumerator IEnumerable.GetEnumerator() { return (IEnumerator)GetEnumerator(); }
}
```
The above code runs in about three quarters of the time as the above Priority Queue based version for a range of a million primes which will fall even further behind for increasing ranges due to the Dictionary providing O(1) access times as compared to the O(log n) access times for the Priority Queue; the only slight advantage of the PQ based version is at very small ranges where the constant factor overhead of computing the table hashes becomes greater than the "log n" factor for small "n".
'''Page Segmented Array Sieve'''
All of the above unbounded versions are really just an intellectual exercise as with very little extra lines of code above the fastest Dictionary based version, one can have an bit-packed page-segmented array based version as follows:
```c#
using System;
using System.Collections;
using System.Collections.Generic;
using System.Linq;
using PrimeT = System.UInt32;
class PrimesPgd : IEnumerable {
private const int PGSZ = 1 << 14; // L1 CPU cache size in bytes
private const int BFBTS = PGSZ * 8; // in bits
private const int BFRNG = BFBTS * 2;
public IEnumerator nmrtr() {
IEnumerator bps = null;
List bpa = new List();
uint[] cbuf = new uint[PGSZ / 4]; // 4 byte words
yield return 2;
for (var lowi = (PrimeT)0; ; lowi += BFBTS) {
for (var bi = 0; ; ++bi) {
if (bi < 1) {
if (bi < 0) { bi = 0; yield return 2; }
PrimeT nxt = 3 + lowi + lowi + BFRNG;
if (lowi <= 0) { // cull very first page
for (int i = 0, p = 3, sqr = 9; sqr < nxt; i++, p += 2, sqr = p * p)
if ((cbuf[i >> 5] & (1 << (i & 31))) == 0)
for (int j = (sqr - 3) >> 1; j < BFBTS; j += p)
cbuf[j >> 5] |= 1u << j;
}
else { // cull for the rest of the pages
Array.Clear(cbuf, 0, cbuf.Length);
if (bpa.Count == 0) { // inite secondar base primes stream
bps = nmrtr(); bps.MoveNext(); bps.MoveNext();
bpa.Add((uint)bps.Current); bps.MoveNext();
} // add 3 to base primes array
// make sure bpa contains enough base primes...
for (PrimeT p = bpa[bpa.Count - 1], sqr = p * p; sqr < nxt; ) {
p = bps.Current; bps.MoveNext(); sqr = p * p; bpa.Add((uint)p);
}
for (int i = 0, lmt = bpa.Count - 1; i < lmt; i++) {
var p = (PrimeT)bpa[i]; var s = (p * p - 3) >> 1;
// adjust start index based on page lower limit...
if (s >= lowi) s -= lowi;
else {
var r = (lowi - s) % p;
s = (r != 0) ? p - r : 0;
}
for (var j = (uint)s; j < BFBTS; j += p)
cbuf[j >> 5] |= 1u << ((int)j);
}
}
}
while (bi < BFBTS && (cbuf[bi >> 5] & (1 << (bi & 31))) != 0) ++bi;
if (bi < BFBTS) yield return 3 + (((PrimeT)bi + lowi) << 1);
else break; // outer loop for next page segment...
}
}
}
public IEnumerator GetEnumerator() { return nmrtr(); }
IEnumerator IEnumerable.GetEnumerator() { return (IEnumerator)GetEnumerator(); }
}
```
The above code is about 25 times faster than the Dictionary version at computing the first about 50 million primes (up to a range of one billion), with the actual enumeration of the result sequence now taking longer than the time it takes to cull the composite number representation bits from the arrays, meaning that it is over 50 times faster at actually sieving the primes. The code owes its speed as compared to a naive "one huge memory array" algorithm to using an array size that is the size of the CPU L1 or L2 caches and using bit-packing to fit more number representations into this limited capacity; in this way RAM memory access times are reduced by a factor of from about four to about 10 (depending on CPU and RAM speed) as compared to those naive implementations, and the minor computational cost of the bit manipulations is compensated by a large factor in total execution time.
The time to enumerate the result primes sequence can be reduced somewhat (about a second) by removing the automatic iterator "yield return" statements and converting them into a "rull-your-own" IEnumerable implementation, but for page segmentation of '''odds-only''', this iteration of the results will still take longer than the time to actually cull the composite numbers from the page arrays.
In order to make further gains in speed, custom methods must be used to avoid using iterator sequences. If this is done, then further gains can be made by extreme wheel factorization (up to about another about four times gain in speed) and multi-processing (with another gain in speed proportional to the actual independent CPU cores used).
Note that all of these gains in speed are not due to C# other than it compiles to reasonably efficient machine code, but rather to proper use of the Sieve of Eratosthenes algorithm.
All of the above unbounded code can be tested by the following "main" method (replace the name "PrimesXXX" with the name of the class to be tested):
```c#
static void Main(string[] args) {
Console.WriteLine(PrimesXXX().ElementAt(1000000 - 1)); // zero based indexing...
}
```
To produce the following output for all tested versions (although some are considerably faster than others):
```txt
15485863
```
## Chapel
This solution uses nested iterators to create new wheels at run time:
```chapel
// yield prime and remove all multiples of it from children sieves
iter sieve(prime):int {
yield prime;
var last = prime;
label candidates for candidate in sieve(prime+1) do {
for composite in last..candidate by prime do {
// candidate is a multiple of this prime
if composite == candidate then {
// remember size of last composite
last = composite;
// and try the next candidate
continue candidates;
}
}
// candidate cannot need to be removed by this sieve
// yield to parent sieve for checking
yield candidate;
}
}
```
The topmost sieve needs to be started with 2 (the smallest prime):
```chapel
config const N = 30;
for p in sieve(2) {
write(" ", p);
if p > N then {
writeln();
break;
}
}
```
## Clojure
Note: this is not a Sieve of Eratosthenes; it is just an inefficient version (because it doesn't trial just the found primes <= the square root of the range) trial division.
''primes<'' is a functional interpretation of the Sieve of Eratosthenes. It merely removes the set of composite numbers from the set of odd numbers (wheel of 2) leaving behind only prime numbers. It uses a transducer internally with "into #{}".
```clojure
(defn primes< [n]
(if (<= n 2)
()
(remove (into #{}
(mapcat #(range (* % %) n %))
(range 3 (Math/sqrt n) 2))
(cons 2 (range 3 n 2)))))
```
Calculates primes up to and including ''n'' using a mutable boolean array but otherwise entirely functional code.
```clojure
(defn primes-to
"Computes lazy sequence of prime numbers up to a given number using sieve of Eratosthenes"
[n]
(let [root (-> n Math/sqrt long),
cmpsts (boolean-array (inc n)),
cullp (fn [p]
(loop [i (* p p)]
(if (<= i n)
(do (aset cmpsts i true)
(recur (+ i p))))))]
(do (dorun (map #(cullp %) (filter #(not (aget cmpsts %))
(range 2 (inc root)))))
(filter #(not (aget cmpsts %)) (range 2 (inc n))))))
```
'''Alternative implementation using Clojure's side-effect oriented list comprehension.'''
```clojure
(defn primes-to
"Returns a lazy sequence of prime numbers less than lim"
[lim]
(let [refs (boolean-array (+ lim 1) true)
root (int (Math/sqrt lim))]
(do (doseq [i (range 2 lim)
:while (<= i root)
:when (aget refs i)]
(doseq [j (range (* i i) lim i)]
(aset refs j false)))
(filter #(aget refs %) (range 2 lim)))))
```
'''Alternative implementation using Clojure's side-effect oriented list comprehension. Odds only.'''
```clojure
(defn primes-to
"Returns a lazy sequence of prime numbers less than lim"
[lim]
(let [max-i (int (/ (- lim 1) 2))
refs (boolean-array max-i true)
root (/ (dec (int (Math/sqrt lim))) 2)]
(do (doseq [i (range 1 (inc root))
:when (aget refs i)]
(doseq [j (range (* (+ i i) (inc i)) max-i (+ i i 1))]
(aset refs j false)))
(cons 2 (map #(+ % % 1) (filter #(aget refs %) (range 1 max-i)))))))
```
This implemantation is about twice fast than previous one and use only half memory.
From the index of array calculates the value it represents as (2*i + 1), the step between two index that represents
the multiples of primes to mark as composite is also (2*i + 1).
The index of the square of the prime to start composite marking is 2*i*(i+1).
'''Alternative very slow entirely functional implementation using lazy sequences'''
```clojure
(defn primes-to
"Computes lazy sequence of prime numbers up to a given number using sieve of Eratosthenes"
[n]
(letfn [(nxtprm [cs] ; current candidates
(let [p (first cs)]
(if (> p (Math/sqrt n)) cs
(cons p (lazy-seq (nxtprm (-> (range (* p p) (inc n) p)
set (remove cs) rest)))))))]
(nxtprm (range 2 (inc n)))))
```
The reason that the above code is so slow is that it has has a high constant factor overhead due to using a (hash) set to remove the composites from the future composites stream, each prime composite stream removal requires a scan across all remaining composites (compared to using an array or vector where only the culled values are referenced, and due to the slowness of Clojure sequence operations as compared to iterator/sequence operations in other languages.
'''Version based on immutable Vector's'''
Here is an immutable boolean vector based non-lazy sequence version other than for the lazy sequence operations to output the result:
```clojure
(defn primes-to
"Computes lazy sequence of prime numbers up to a given number using sieve of Eratosthenes"
[max-prime]
(let [sieve (fn [s n]
(if (<= (* n n) max-prime)
(recur (if (s n)
(reduce #(assoc %1 %2 false) s (range (* n n) (inc max-prime) n))
s)
(inc n))
s))]
(->> (-> (reduce conj (vector-of :boolean) (map #(= % %) (range (inc max-prime))))
(assoc 0 false)
(assoc 1 false)
(sieve 2))
(map-indexed #(vector %2 %1)) (filter first) (map second))))
```
The above code is still quite slow due to the cost of the immutable copy-on-modify operations.
'''Odds only bit packed mutable array based version'''
The following code implements an odds-only sieve using a mutable bit packed long array, only using a lazy sequence for the output of the resulting primes:
```clojure
(set! *unchecked-math* true)
(defn primes-to
"Computes lazy sequence of prime numbers up to a given number using sieve of Eratosthenes"
[n]
(let [root (-> n Math/sqrt long),
rootndx (long (/ (- root 3) 2)),
ndx (long (/ (- n 3) 2)),
cmpsts (long-array (inc (/ ndx 64))),
isprm #(zero? (bit-and (aget cmpsts (bit-shift-right % 6))
(bit-shift-left 1 (bit-and % 63)))),
cullp (fn [i]
(let [p (long (+ i i 3))]
(loop [i (bit-shift-right (- (* p p) 3) 1)]
(if (<= i ndx)
(do (let [w (bit-shift-right i 6)]
(aset cmpsts w (bit-or (aget cmpsts w)
(bit-shift-left 1 (bit-and i 63)))))
(recur (+ i p))))))),
cull (fn [] (loop [i 0] (if (<= i rootndx)
(do (if (isprm i) (cullp i)) (recur (inc i))))))]
(letfn [(nxtprm [i] (if (<= i ndx)
(cons (+ i i 3) (lazy-seq (nxtprm (loop [i (inc i)]
(if (or (> i ndx) (isprm i)) i
(recur (inc i)))))))))]
(if (< n 2) nil
(cons 3 (if (< n 3) nil (do (cull) (lazy-seq (nxtprm 0)))))))))
```
The above code is about as fast as any "one large sieving array" type of program in any computer language with this level of wheel factorization other than the lazy sequence operations are quite slow: it takes about ten times as long to enumerate the results as it does to do the actual sieving work of culling the composites from the sieving buffer array. The slowness of sequence operations is due to nested function calls, but primarily due to the way Clojure implements closures by "boxing" all arguments (and perhaps return values) as objects in the heap space, which then need to be "un-boxed" as primitives as necessary for integer operations. Some of the facilities provided by lazy sequences are not needed for this algorithm, such as the automatic memoization which means that each element of the sequence is calculated only once; it is not necessary for the sequence values to be retraced for this algorithm.
If further levels of wheel factorization were used, the time to enumerate the resulting primes would be an even higher overhead as compared to the actual composite number culling time, would get even higher if page segmentation were used to limit the buffer size to the size of the CPU L1 cache for many times better memory access times, most important in the culling operations, and yet higher again if multi-processing were used to share to page segment processing across CPU cores.
The following code overcomes many of those limitations by using an internal (OPSeq) "deftype" which implements the ISeq interface as well as the Counted interface to provide immediate count returns (based on a pre-computed total), as well as the IReduce interface which can greatly speed come computations based on the primes sequence (eased greatly using facilities provided by Clojure 1.7.0 and up):
```clojure
(defn primes-tox
"Computes lazy sequence of prime numbers up to a given number using sieve of Eratosthenes"
[n]
(let [root (-> n Math/sqrt long),
rootndx (long (/ (- root 3) 2)),
ndx (max (long (/ (- n 3) 2)) 0),
lmt (quot ndx 64),
cmpsts (long-array (inc lmt)),
cullp (fn [i]
(let [p (long (+ i i 3))]
(loop [i (bit-shift-right (- (* p p) 3) 1)]
(if (<= i ndx)
(do (let [w (bit-shift-right i 6)]
(aset cmpsts w (bit-or (aget cmpsts w)
(bit-shift-left 1 (bit-and i 63)))))
(recur (+ i p))))))),
cull (fn [] (do (aset cmpsts lmt (bit-or (aget cmpsts lmt)
(bit-shift-left -2 (bit-and ndx 63))))
(loop [i 0]
(when (<= i rootndx)
(when (zero? (bit-and (aget cmpsts (bit-shift-right i 6))
(bit-shift-left 1 (bit-and i 63))))
(cullp i))
(recur (inc i))))))
numprms (fn []
(let [w (dec (alength cmpsts))] ;; fast results count bit counter
(loop [i 0, cnt (bit-shift-left (alength cmpsts) 6)]
(if (> i w) cnt
(recur (inc i)
(- cnt (java.lang.Long/bitCount (aget cmpsts i))))))))]
(if (< n 2) nil
(cons 2 (if (< n 3) nil
(do (cull)
(deftype OPSeq [^long i ^longs cmpsa ^long cnt ^long tcnt] ;; for arrays maybe need to embed the array so that it doesn't get garbage collected???
clojure.lang.ISeq
(first [_] (if (nil? cmpsa) nil (+ i i 3)))
(next [_] (let [ncnt (inc cnt)] (if (>= ncnt tcnt) nil
(OPSeq.
(loop [j (inc i)]
(let [p? (zero? (bit-and (aget cmpsa (bit-shift-right j 6))
(bit-shift-left 1 (bit-and j 63))))]
(if p? j (recur (inc j)))))
cmpsa ncnt tcnt))))
(more [this] (let [ncnt (inc cnt)] (if (>= ncnt tcnt) (OPSeq. 0 nil tcnt tcnt)
(.next this))))
(cons [this o] (clojure.core/cons o this))
(empty [_] (if (= cnt tcnt) nil (OPSeq. 0 nil tcnt tcnt)))
(equiv [this o] (if (or (not= (type this) (type o))
(not= cnt (.cnt ^OPSeq o)) (not= tcnt (.tcnt ^OPSeq o))
(not= i (.i ^OPSeq o))) false true))
clojure.lang.Counted
(count [_] (- tcnt cnt))
clojure.lang.Seqable
(clojure.lang.Seqable/seq [this] (if (= cnt tcnt) nil this))
clojure.lang.IReduce
(reduce [_ f v] (let [c (- tcnt cnt)]
(if (<= c 0) nil
(loop [ci i, n c, rslt v]
(if (zero? (bit-and (aget cmpsa (bit-shift-right ci 6))
(bit-shift-left 1 (bit-and ci 63))))
(let [rrslt (f rslt (+ ci ci 3)),
rdcd (reduced? rrslt),
nrslt (if rdcd @rrslt rrslt)]
(if (or (<= n 1) rdcd) nrslt
(recur (inc ci) (dec n) nrslt)))
(recur (inc ci) n rslt))))))
(reduce [this f] (if (nil? i) (f) (if (= (.count this) 1) (+ i i 3)
(.reduce ^clojure.lang.IReduce (.next this) f (+ i i 3)))))
clojure.lang.Sequential
Object
(toString [this] (if (= cnt tcnt) "()"
(.toString (seq (map identity this))))))
(->OPSeq 0 cmpsts 0 (numprms))))))))
```
'(time (count (primes-tox 10000000)))' takes about 40 milliseconds (compiled) to produce 664579.
Due to the better efficiency of the custom CIS type, the primes to the above range can be enumerated in about the same 40 milliseconds that it takes to cull and count the sieve buffer array.
Under Clojure 1.7.0, one can use '(time (reduce (fn [] (+ (long sum) (long v))) 0 (primes-tox 2000000)))' to find "142913828922" as the sum of the primes to two million as per [https://projecteuler.net/problem=10 Euler Problem 10] in about 40 milliseconds total with about half the time used for sieving the array and half for computing the sum.
To show how sensitive Clojure is to forms of expression of functions, the simple form '(time (reduce + (primes-tox 2000000)))' takes about twice as long even though it is using the same internal routine for most of the calculation due to the function not having the type coercion's.
Before one considers that this code is suitable for larger ranges, it is still lacks the improvements of page segmentation with pages about the size of the CPU L1/L2 caches (produces about a four times speed up), maximal wheel factorization (to make it another about four times faster), and the use of multi-processing (for a further gain of about 4 times for a multi-core desktop CPU such as an Intel i7), will make the sieving/counting code about 50 times faster than this, although there will only be a moderate improvement in the time to enumerate/process the resulting primes. Using these techniques, the number of primes to one billion can be counted in a small fraction of a second.
### Unbounded Versions
For some types of problems such as finding the nth prime (rather than the sequence of primes up to m), a prime sieve with no upper bound is a better tool.
The following variations on an incremental Sieve of Eratosthenes are based on or derived from the Richard Bird sieve as described in the Epilogue of [http://www.cs.hmc.edu/~oneill/papers/Sieve-JFP.pdf Melissa E. O'Neill's definitive paper]:
'''A Clojure version of Richard Bird's Sieve using Lazy Sequences (sieves odds only)'''
```clojure
(defn primes-Bird
"Computes the unbounded sequence of primes using a Sieve of Eratosthenes algorithm by Richard Bird."
[]
(letfn [(mltpls [p] (let [p2 (* 2 p)]
(letfn [(nxtmltpl [c]
(cons c (lazy-seq (nxtmltpl (+ c p2)))))]
(nxtmltpl (* p p))))),
(allmtpls [ps] (cons (mltpls (first ps)) (lazy-seq (allmtpls (next ps))))),
(union [xs ys] (let [xv (first xs), yv (first ys)]
(if (< xv yv) (cons xv (lazy-seq (union (next xs) ys)))
(if (< yv xv) (cons yv (lazy-seq (union xs (next ys))))
(cons xv (lazy-seq (union (next xs) (next ys)))))))),
(mrgmltpls [mltplss] (cons (first (first mltplss))
(lazy-seq (union (next (first mltplss))
(mrgmltpls (next mltplss)))))),
(minusStrtAt [n cmpsts] (loop [n n, cmpsts cmpsts]
(if (< n (first cmpsts))
(cons n (lazy-seq (minusStrtAt (+ n 2) cmpsts)))
(recur (+ n 2) (next cmpsts)))))]
(do (def oddprms (cons 3 (lazy-seq (let [cmpsts (-> oddprms (allmtpls) (mrgmltpls))]
(minusStrtAt 5 cmpsts)))))
(cons 2 (lazy-seq oddprms)))))
```
The above code is quite slow due to both that the data structure is a linear merging of prime multiples and due to the slowness of the Clojure sequence operations.
'''A Clojure version of the tree folding sieve using Lazy Sequences'''
The following code speeds up the above code by merging the linear sequence of sequences as above by pairs into a right-leaning tree structure:
```clojure
(defn primes-treeFolding
"Computes the unbounded sequence of primes using a Sieve of Eratosthenes algorithm modified from Bird."
[]
(letfn [(mltpls [p] (let [p2 (* 2 p)]
(letfn [(nxtmltpl [c]
(cons c (lazy-seq (nxtmltpl (+ c p2)))))]
(nxtmltpl (* p p))))),
(allmtpls [ps] (cons (mltpls (first ps)) (lazy-seq (allmtpls (next ps))))),
(union [xs ys] (let [xv (first xs), yv (first ys)]
(if (< xv yv) (cons xv (lazy-seq (union (next xs) ys)))
(if (< yv xv) (cons yv (lazy-seq (union xs (next ys))))
(cons xv (lazy-seq (union (next xs) (next ys)))))))),
(pairs [mltplss] (let [tl (next mltplss)]
(cons (union (first mltplss) (first tl))
(lazy-seq (pairs (next tl)))))),
(mrgmltpls [mltplss] (cons (first (first mltplss))
(lazy-seq (union (next (first mltplss))
(mrgmltpls (pairs (next mltplss))))))),
(minusStrtAt [n cmpsts] (loop [n n, cmpsts cmpsts]
(if (< n (first cmpsts))
(cons n (lazy-seq (minusStrtAt (+ n 2) cmpsts)))
(recur (+ n 2) (next cmpsts)))))]
(do (def oddprms (cons 3 (lazy-seq (let [cmpsts (-> oddprms (allmtpls) (mrgmltpls))]
(minusStrtAt 5 cmpsts)))))
(cons 2 (lazy-seq oddprms)))))
```
The above code is still slower than it should be due to the slowness of Clojure's sequence operations.
'''A Clojure version of the above tree folding sieve using a custom Co Inductive Sequence'''
The following code uses a custom "deftype" non-memoizing Co Inductive Stream/Sequence (CIS) implementing the ISeq interface to make the sequence operations more efficient and is about four times faster than the above code:
```clojure
(deftype CIS [v cont]
clojure.lang.ISeq
(first [_] v)
(next [_] (if (nil? cont) nil (cont)))
(more [this] (let [nv (.next this)] (if (nil? nv) (CIS. nil nil) nv)))
(cons [this o] (clojure.core/cons o this))
(empty [_] (if (and (nil? v) (nil? cont)) nil (CIS. nil nil)))
(equiv [this o] (loop [cis1 this, cis2 o] (if (nil? cis1) (if (nil? cis2) true false)
(if (or (not= (type cis1) (type cis2))
(not= (.v cis1) (.v ^CIS cis2))
(and (nil? (.cont cis1))
(not (nil? (.cont ^CIS cis2))))
(and (nil? (.cont ^CIS cis2))
(not (nil? (.cont cis1))))) false
(if (nil? (.cont cis1)) true
(recur ((.cont cis1)) ((.cont ^CIS cis2))))))))
(count [this] (loop [cis this, cnt 0] (if (or (nil? cis) (nil? (.cont cis))) cnt
(recur ((.cont cis)) (inc cnt)))))
clojure.lang.Seqable
(seq [this] (if (and (nil? v) (nil? cont)) nil this))
clojure.lang.Sequential
Object
(toString [this] (if (and (nil? v) (nil? cont)) "()" (.toString (seq (map identity this))))))
(defn primes-treeFoldingx
"Computes the unbounded sequence of primes using a Sieve of Eratosthenes algorithm modified from Bird."
[]
(letfn [(mltpls [p] (let [p2 (* 2 p)]
(letfn [(nxtmltpl [c]
(->CIS c (fn [] (nxtmltpl (+ c p2)))))]
(nxtmltpl (* p p))))),
(allmtpls [^CIS ps] (->CIS (mltpls (.v ps)) (fn [] (allmtpls ((.cont ps)))))),
(union [^CIS xs ^CIS ys] (let [xv (.v xs), yv (.v ys)]
(if (< xv yv) (->CIS xv (fn [] (union ((.cont xs)) ys)))
(if (< yv xv) (->CIS yv (fn [] (union xs ((.cont ys)))))
(->CIS xv (fn [] (union (next xs) ((.cont ys))))))))),
(pairs [^CIS mltplss] (let [^CIS tl ((.cont mltplss))]
(->CIS (union (.v mltplss) (.v tl))
(fn [] (pairs ((.cont tl))))))),
(mrgmltpls [^CIS mltplss] (->CIS (.v ^CIS (.v mltplss))
(fn [] (union ((.cont ^CIS (.v mltplss)))
(mrgmltpls (pairs ((.cont mltplss)))))))),
(minusStrtAt [n ^CIS cmpsts] (loop [n n, cmpsts cmpsts]
(if (< n (.v cmpsts))
(->CIS n (fn [] (minusStrtAt (+ n 2) cmpsts)))
(recur (+ n 2) ((.cont cmpsts))))))]
(do (def oddprms (->CIS 3 (fn [] (let [cmpsts (-> oddprms (allmtpls) (mrgmltpls))]
(minusStrtAt 5 cmpsts)))))
(->CIS 2 (fn [] oddprms)))))
```
'(time (count (take-while #(<= (long %) 10000000) (primes-treeFoldingx))))' takes about 3.4 seconds for a range of 10 million.
The above code is useful for ranges up to about fifteen million primes, which is about the first million primes; it is comparable in speed to all of the bounded versions except for the fastest bit packed version which can reasonably be used for ranges about 100 times as large.
'''Incremental Hash Map based unbounded "odds-only" version'''
The following code is a version of the O'Neill Haskell code but does not use wheel factorization other than for sieving odds only (although it could be easily added) and uses a Hash Map (constant amortized access time) rather than a Priority Queue (log n access time for combined remove-and-insert-anew operations, which are the majority used for this algorithm) with a lazy sequence for output of the resulting primes; the code has the added feature that it uses a secondary base primes sequence generator and only adds prime culling sequences to the composites map when they are necessary, thus saving time and limiting storage to only that required for the map entries for primes up to the square root of the currently sieved number:
```clojure
(defn primes-hashmap
"Infinite sequence of primes using an incremental Sieve or Eratosthenes with a Hashmap"
[]
(letfn [(nxtoddprm [c q bsprms cmpsts]
(if (>= c q) ;; only ever equal
(let [p2 (* (first bsprms) 2), nbps (next bsprms), nbp (first nbps)]
(recur (+ c 2) (* nbp nbp) nbps (assoc cmpsts (+ q p2) p2)))
(if (contains? cmpsts c)
(recur (+ c 2) q bsprms
(let [adv (cmpsts c), ncmps (dissoc cmpsts c)]
(assoc ncmps
(loop [try (+ c adv)] ;; ensure map entry is unique
(if (contains? ncmps try)
(recur (+ try adv)) try)) adv)))
(cons c (lazy-seq (nxtoddprm (+ c 2) q bsprms cmpsts))))))]
(do (def baseoddprms (cons 3 (lazy-seq (nxtoddprm 5 9 baseoddprms {}))))
(cons 2 (lazy-seq (nxtoddprm 3 9 baseoddprms {}))))))
```
The above code is slower than the best tree folding version due to the added constant factor overhead of computing the hash functions for every hash map operation even though it has computational complexity of (n log log n) rather than the worse (n log n log log n) for the previous incremental tree folding sieve. It is still about 100 times slower than the sieve based on the bit-packed mutable array due to these constant factor hashing overheads.
There is almost no benefit of converting the above code to use a CIS as most of the time is expended in the hash map functions.
'''Incremental Priority Queue based unbounded "odds-only" version'''
In order to implement the O'Neill Priority Queue incremental Sieve of Eratosthenes algorithm, one requires an efficient implementation of a Priority Queue, which is not part of standard Clojure. For this purpose, the most suitable Priority Queue is a binary tree heap based MinHeap algorithm. The following code implements a purely functional (using entirely immutable state) MinHeap Priority Queue providing the required functions of (emtpy-pq) initialization, (getMin-pq pq) to examinte the minimum key/value pair in the queue, (insert-pq pq k v) to add entries to the queue, and (replaceMinAs-pq pq k v) to replaace the minimum entry with a key/value pair as given (it is more efficient that if functions were provided to delete and then re-insert entries in the queue; there is therefore no "delete" or other queue functions supplied as the algorithm does not requrie them:
```clojure
(deftype PQEntry [k, v]
Object
(toString [_] (str "<" k "," v ">")))
(deftype PQNode [ntry, lft, rght]
Object
(toString [_] (str "<" ntry " left: " (str lft) " right: " (str rght) ">")))
(defn empty-pq [] nil)
(defn getMin-pq [^PQNode pq]
(if (nil? pq)
nil
(.ntry pq)))
(defn insert-pq [^PQNode opq ok v]
(loop [^PQEntry kv (->PQEntry ok v), pq opq, cont identity]
(if (nil? pq)
(cont (->PQNode kv nil nil))
(let [k (.k kv),
^PQEntry kvn (.ntry pq), kn (.k kvn),
l (.lft pq), r (.rght pq)]
(if (<= k kn)
(recur kvn r #(cont (->PQNode kv % l)))
(recur kv r #(cont (->PQNode kvn % l))))))))
(defn replaceMinAs-pq [^PQNode opq k v]
(let [^PQEntry kv (->PQEntry k v)]
(if (nil? opq) ;; if was empty or just an entry, just use current entry
(->PQNode kv nil nil)
(loop [pq opq, cont identity]
(let [^PQNode l (.lft pq), ^PQNode r (.rght pq)]
(cond ;; if left us empty, right must be too
(nil? l)
(cont (->PQNode kv nil nil)),
(nil? r) ;; we only have a left...
(let [^PQEntry kvl (.ntry l), kl (.k kvl)]
(if (<= k kl)
(cont (->PQNode kv l nil))
(recur l #(cont (->PQNode kvl % nil))))),
:else (let [^PQEntry kvl (.ntry l), kl (.k kvl),
^PQEntry kvr (.ntry r), kr (.k kvr)] ;; we have both
(if (and (<= k kl) (<= k kr))
(cont (->PQNode kv l r))
(if (<= kl kr)
(recur l #(cont (->PQNode kvl % r)))
(recur r #(cont (->PQNode kvr l %))))))))))))
```
Note that the above code is written partially using continuation passing style so as to leave the "recur" calls in tail call position as required for efficient looping in Clojure; for practical sieving ranges, the algorithm could likely use just raw function recursion as recursion depth is unlikely to be used beyond a depth of about ten or so, but raw recursion is said to be less code efficient.
The actual incremental sieve using the Priority Queue is as follows, which code uses the same optimizations of postponing the addition of prime composite streams to the queue until the square root of the currently sieved number is reached and using a secondary base primes stream to generate the primes composite stream markers in the queue as was used for the Hash Map version:
```clojure
(defn primes-pq
"Infinite sequence of primes using an incremental Sieve or Eratosthenes with a Priority Queue"
[]
(letfn [(nxtoddprm [c q bsprms cmpsts]
(if (>= c q) ;; only ever equal
(let [p2 (* (first bsprms) 2), nbps (next bsprms), nbp (first nbps)]
(recur (+ c 2) (* nbp nbp) nbps (insert-pq cmpsts (+ q p2) p2)))
(let [mn (getMin-pq cmpsts)]
(if (and mn (>= c (.k mn))) ;; never greater than
(recur (+ c 2) q bsprms
(loop [adv (.v mn), cmps cmpsts] ;; advance repeat composites for value
(let [ncmps (replaceMinAs-pq cmps (+ c adv) adv),
nmn (getMin-pq ncmps)]
(if (and nmn (>= c (.k nmn)))
(recur (.v nmn) ncmps)
ncmps))))
(cons c (lazy-seq (nxtoddprm (+ c 2) q bsprms cmpsts)))))))]
(do (def baseoddprms (cons 3 (lazy-seq (nxtoddprm 5 9 baseoddprms (empty-pq)))))
(cons 2 (lazy-seq (nxtoddprm 3 9 baseoddprms (empty-pq)))))))
```
The above code is faster than the Hash Map version up to about a sieving range of fifteen million or so, but gets progressively slower for larger ranges due to having (n log n log log n) computational complexity rather than the (n log log n) for the Hash Map version, which has a higher constant factor overhead that is overtaken by the extra "log n" factor.
It is slower that the fastest of the tree folding versions (which has the same computational complexity) due to the higher constant factor overhead of the Priority Queue operations (although perhaps a more efficient implementation of the MinHeap Priority Queue could be developed).
Again, these non-mutable array based sieves are about a hundred times slower than even the "one large memory buffer array" version as implemented in the bounded section; a page segmented version of the mutable bit-packed memory array would be several times faster.
All of these algorithms will respond to maximum wheel factorization, getting up to approximately four times faster if this is applied as compared to the the "odds-only" versions.
It is difficult if not impossible to apply efficient multi-processing to the above versions of the unbounded sieves as the next values of the primes sequence are dependent on previous changes of state for the Bird and Tree Folding versions; however, with the addition of a "update the whole Priority Queue (and reheapify)" or "update the Hash Map" to a given page start state functions, it is possible to do for these letter two algorithms; however, even though it is possible and there is some benefit for these latter two implementations, the benefit is less than using mutable arrays due to that the results must be enumerated into a data structure of some sort in order to be passed out of the page function whereas they can be directly enumerated from the array for the mutable array versions.
'''Bit packed page segmented array unbounded "odds-only" version'''
To show that Clojure does not need to be particularly slow, the following version runs about twice as fast as the non-segmented unbounded array based version above (extremely fast compared to the non-array based versions) and only a little slower than other equivalent versions running on virtual machines: C# or F# on DotNet or Java and Scala on the JVM:
```clojure
(set! *unchecked-math* true)
(def PGSZ (bit-shift-left 1 14)) ;; size of CPU cache
(def PGBTS (bit-shift-left PGSZ 3))
(def PGWRDS (bit-shift-right PGBTS 5))
(def BPWRDS (bit-shift-left 1 7)) ;; smaller page buffer for base primes
(def BPBTS (bit-shift-left BPWRDS 5))
(defn- count-pg
"count primes in the culled page buffer, with test for limit"
[lmt ^ints pg]
(let [pgsz (alength pg),
pgbts (bit-shift-left pgsz 5),
cntem (fn [lmtw]
(let [lmtw (long lmtw)]
(loop [i (long 0), c (long 0)]
(if (>= i lmtw) (- (bit-shift-left lmtw 5) c)
(recur (inc i)
(+ c (java.lang.Integer/bitCount (aget pg i))))))))]
(if (< lmt pgbts)
(let [lmtw (bit-shift-right lmt 5),
lmtb (bit-and lmt 31)
msk (bit-shift-left -2 lmtb)]
(+ (cntem lmtw)
(- 32 (java.lang.Integer/bitCount (bit-or (aget pg lmtw)
msk)))))
(- pgbts
(areduce pg i ret (long 0) (+ ret (java.lang.Integer/bitCount (aget pg i))))))))
;; (cntem pgsz))))
(defn- primes-pages
"unbounded Sieve of Eratosthenes producing a lazy sequence of culled page buffers."
[]
(letfn [(make-pg [lowi pgsz bpgs]
(let [lowi (long lowi),
pgbts (long (bit-shift-left pgsz 5)),
pgrng (long (+ (bit-shift-left (+ lowi pgbts) 1) 3)),
^ints pg (int-array pgsz),
cull (fn [bpgs']
(loop [i (long 0), bpgs' bpgs']
(let [^ints fbpg (first bpgs'),
bpgsz (long (alength fbpg))]
(if (>= i bpgsz)
(recur 0 (next bpgs'))
(let [p (long (aget fbpg i)),
sqr (long (* p p))]
(if (< sqr pgrng) (do
(loop [j (long (let [s (long (bit-shift-right (- sqr 3) 1))]
(if (>= s lowi) (- s lowi)
(let [m (long (rem (- lowi s) p))]
(if (zero? m)
0
(- p m))))))]
(if (< j pgbts) ;; fast inner culling loop where most time is spent
(do
(let [w (bit-shift-right j 5)]
(aset pg w (int (bit-or (aget pg w)
(bit-shift-left 1 (bit-and j 31))))))
(recur (+ j p)))))
(recur (inc i) bpgs'))))))))]
(do (if (nil? bpgs)
(letfn [(mkbpps [i]
(if (zero? (bit-and (aget pg (bit-shift-right i 5))
(bit-shift-left 1 (bit-and i 31))))
(cons (int-array 1 (+ i i 3)) (lazy-seq (mkbpps (inc i))))
(recur (inc i))))]
(cull (mkbpps 0)))
(cull bpgs))
pg))),
(page-seq [lowi pgsz bps]
(letfn [(next-seq [lwi]
(cons (make-pg lwi pgsz bps)
(lazy-seq (next-seq (+ lwi (bit-shift-left pgsz 5))))))]
(next-seq lowi)))
(pgs->bppgs [ppgs]
(letfn [(nxt-pg [lowi pgs]
(let [^ints pg (first pgs),
cnt (count-pg BPBTS pg),
npg (int-array cnt)]
(do (loop [i 0, j 0]
(if (< i BPBTS)
(if (zero? (bit-and (aget pg (bit-shift-right i 5))
(bit-shift-left 1 (bit-and i 31))))
(do (aset npg j (+ (bit-shift-left (+ lowi i) 1) 3))
(recur (inc i) (inc j)))
(recur (inc i) j))))
(cons npg (lazy-seq (nxt-pg (+ lowi BPBTS) (next pgs)))))))]
(nxt-pg 0 ppgs))),
(make-base-prms-pgs []
(pgs->bppgs (cons (make-pg 0 BPWRDS nil)
(lazy-seq (page-seq BPBTS BPWRDS (make-base-prms-pgs))))))]
(page-seq 0 PGWRDS (make-base-prms-pgs))))
(defn primes-paged
"unbounded Sieve of Eratosthenes producing a lazy sequence of primes"
[]
(do (deftype CIS [v cont]
clojure.lang.ISeq
(first [_] v)
(next [_] (if (nil? cont) nil (cont)))
(more [this] (let [nv (.next this)] (if (nil? nv) (CIS. nil nil) nv)))
(cons [this o] (clojure.core/cons o this))
(empty [_] (if (and (nil? v) (nil? cont)) nil (CIS. nil nil)))
(equiv [this o] (loop [cis1 this, cis2 o] (if (nil? cis1) (if (nil? cis2) true false)
(if (or (not= (type cis1) (type cis2))
(not= (.v cis1) (.v ^CIS cis2))
(and (nil? (.cont cis1))
(not (nil? (.cont ^CIS cis2))))
(and (nil? (.cont ^CIS cis2))
(not (nil? (.cont cis1))))) false
(if (nil? (.cont cis1)) true
(recur ((.cont cis1)) ((.cont ^CIS cis2))))))))
(count [this] (loop [cis this, cnt 0] (if (or (nil? cis) (nil? (.cont cis))) cnt
(recur ((.cont cis)) (inc cnt)))))
clojure.lang.Seqable
(seq [this] (if (and (nil? v) (nil? cont)) nil this))
clojure.lang.Sequential
Object
(toString [this] (if (and (nil? v) (nil? cont)) "()" (.toString (seq (map identity this))))))
(letfn [(next-prm [lowi i pgseq]
(let [lowi (long lowi),
i (long i),
^ints pg (first pgseq),
pgsz (long (alength pg)),
pgbts (long (bit-shift-left pgsz 5)),
ni (long (loop [j (long i)]
(if (or (>= j pgbts)
(zero? (bit-and (aget pg (bit-shift-right j 5))
(bit-shift-left 1 (bit-and j 31)))))
j
(recur (inc j)))))]
(if (>= ni pgbts)
(recur (+ lowi pgbts) 0 (next pgseq))
(->CIS (+ (bit-shift-left (+ lowi ni) 1) 3)
(fn [] (next-prm lowi (inc ni) pgseq))))))]
(->CIS 2 (fn [] (next-prm 0 0 (primes-pages)))))))
(defn primes-paged-count-to
"counts primes generated by page segments by Sieve of Eratosthenes to the top limit"
[top]
(cond (< top 2) 0
(< top 3) 1
:else (letfn [(nxt-pg [lowi pgseq cnt]
(let [topi (bit-shift-right (- top 3) 1)
nxti (+ lowi PGBTS),
pg (first pgseq)]
(if (> nxti topi)
(+ cnt (count-pg (- topi lowi) pg))
(recur nxti
(next pgseq)
(+ cnt (count-pg PGBTS pg))))))]
(nxt-pg 0 (primes-pages) 1))))
```
The above code runs just as fast as other virtual machine languages when run on a 64-bit JVM; however, when run on a 32-bit JVM it runs almost five times slower. This is likely due to Clojure only using 64-bit integers for integer operations and these operations getting JIT compiled to use library functions to simulate those operations using combined 32-bit operations under a 32-bit JVM whereas direct CPU operations can be used on a 64-bit JVM
Clojure does one thing very slowly, just as here: it enumerates extremely slowly as compared to using a more imperative iteration interface; it helps to use a roll-your-own ISeq interface as here, where enumeration of the primes reduces the time from about four times as long as the composite culling operations for those primes to only about one and a half times as long, although one must also write their own sequence handling functions (can't use "take-while" or "count", for instance) in order to enjoy that benefit. That is why the "primes-paged-count-to" function is provided so it takes a negligible percentage of the time to count the primes over a range as compared to the time for the composite culling operations.
The practical range of the above sieve is about 16 million due to the fixed size of the page buffers; in order to extend the range, a larger page buffer could be used up to the size of the CPU L2 or L3 caches. If a 2^20 buffer were used (one Megabyte, as many modern dexktop CPU's easily have in their L3 cache), then the range would be increased up to about 10^14 at a cost of about a factor of two or three in slower memory accesses per composite culling operation loop. The base primes culling page size is already adequate for this range. One could make the culling page size automatically expand with growing range by about the square root of the current prime range with not too many changes to the code.
As for many implementations of unbounded sieves, the base primes less than the square root of the current range are generated by a secondary generated stream of primes; in this case it is done recursively, so another secondary stream generates the base primes for the base primes and so on down to where the innermost generator has only one page in the stream; this only takes one or two recursions for this type of range.
The base primes culling page size is reduced from the page size for the main primes so that there is less overhead for smaller primes ranges; otherwise excess base primes are generated for fairly small sieve ranges.
## CMake
```cmake
function(eratosthenes var limit)
# Check for integer overflow. With CMake using 32-bit signed integer,
# this check fails when limit > 46340.
if(NOT limit EQUAL 0) # Avoid division by zero.
math(EXPR i "(${limit} * ${limit}) / ${limit}")
if(NOT limit EQUAL ${i})
message(FATAL_ERROR "limit is too large, would cause integer overflow")
endif()
endif()
# Use local variables prime_2, prime_3, ..., prime_${limit} as array.
# Initialize array to y => yes it is prime.
foreach(i RANGE 2 ${limit})
set(prime_${i} y)
endforeach(i)
# Gather a list of prime numbers.
set(list)
foreach(i RANGE 2 ${limit})
if(prime_${i})
# Append this prime to list.
list(APPEND list ${i})
# For each multiple of i, set n => no it is not prime.
# Optimization: start at i squared.
math(EXPR square "${i} * ${i}")
if(NOT square GREATER ${limit}) # Avoid fatal error.
foreach(m RANGE ${square} ${limit} ${i})
set(prime_${m} n)
endforeach(m)
endif()
endif(prime_${i})
endforeach(i)
set(${var} ${list} PARENT_SCOPE)
endfunction(eratosthenes)
```
# Print all prime numbers through 100.
eratosthenes(primes 100)
message(STATUS "${primes}")
## COBOL
```cobol
*> Please ignore the asterisks in the first column of the next comments,
*> which are kludges to get syntax highlighting to work.
IDENTIFICATION DIVISION.
PROGRAM-ID. Sieve-Of-Eratosthenes.
DATA DIVISION.
WORKING-STORAGE SECTION.
01 Max-Number USAGE UNSIGNED-INT.
01 Max-Prime USAGE UNSIGNED-INT.
01 Num-Group.
03 Num-Table PIC X VALUE "P"
OCCURS 1 TO 10000000 TIMES DEPENDING ON Max-Number
INDEXED BY Num-Index.
88 Is-Prime VALUE "P" FALSE "N".
01 Current-Prime USAGE UNSIGNED-INT.
01 I USAGE UNSIGNED-INT.
PROCEDURE DIVISION.
DISPLAY "Enter the limit: " WITH NO ADVANCING
ACCEPT Max-Number
DIVIDE Max-Number BY 2 GIVING Max-Prime
* *> Set Is-Prime of all non-prime numbers to false.
SET Is-Prime (1) TO FALSE
PERFORM UNTIL Max-Prime < Current-Prime
* *> Set current-prime to next prime.
ADD 1 TO Current-Prime
PERFORM VARYING Num-Index FROM Current-Prime BY 1
UNTIL Is-Prime (Num-Index)
END-PERFORM
MOVE Num-Index TO Current-Prime
* *> Set Is-Prime of all multiples of current-prime to
* *> false, starting from current-prime sqaured.
COMPUTE Num-Index = Current-Prime ** 2
PERFORM UNTIL Max-Number < Num-Index
SET Is-Prime (Num-Index) TO FALSE
SET Num-Index UP BY Current-Prime
END-PERFORM
END-PERFORM
* *> Display the prime numbers.
PERFORM VARYING Num-Index FROM 1 BY 1
UNTIL Max-Number < Num-Index
IF Is-Prime (Num-Index)
DISPLAY Num-Index
END-IF
END-PERFORM
GOBACK
.
```
## Common Lisp
```lisp
(defun sieve-of-eratosthenes (maximum)
(loop
with sieve = (make-array (1+ maximum)
:element-type 'bit
:initial-element 0)
for candidate from 2 to maximum
when (zerop (bit sieve candidate))
collect candidate
and do (loop for composite from (expt candidate 2)
to maximum by candidate
do (setf (bit sieve composite) 1))))
```
Working with odds only (above twice speedup), and marking composites only for primes up to the square root of the maximum:
```lisp
(defun sieve-odds (maximum)
"Prime numbers sieve for odd numbers.
Returns a list with all the primes that are less than or equal to maximum."
(loop :with maxi = (ash (1- maximum) -1)
:with stop = (ash (isqrt maximum) -1)
:with sieve = (make-array (1+ maxi) :element-type 'bit :initial-element 0)
:for i :from 1 :to maxi
:for odd-number = (1+ (ash i 1))
:when (zerop (sbit sieve i))
:collect odd-number :into values
:when (<= i stop)
:do (loop :for j :from (* i (1+ i) 2) :to maxi :by odd-number
:do (setf (sbit sieve j) 1))
:finally (return (cons 2 values))))
```
The indexation scheme used here interprets each index i as standing for the value 2i+1. Bit 0 is unused, a small price to pay for the simpler index calculations compared with the 2i+3 indexation scheme. The multiples of a given odd prime p are enumerated in increments of 2p, which corresponds to the index increment of p on the sieve array. The starting point p*p = (2i+1)(2i+1) = 4i(i+1)+1 corresponds to the index 2i(i+1).
While formally a ''wheel'', odds are uniformly spaced and do not require any special processing except for value translation. Wheels proper aren't uniformly spaced and are thus trickier.
## D
### Simpler Version
Prints all numbers less than the limit.
```d
import std.stdio, std.algorithm, std.range, std.functional;
uint[] sieve(in uint limit) nothrow @safe {
if (limit < 2)
return [];
auto composite = new bool[limit];
foreach (immutable uint n; 2 .. cast(uint)(limit ^^ 0.5) + 1)
if (!composite[n])
for (uint k = n * n; k < limit; k += n)
composite[k] = true;
//return iota(2, limit).filter!(not!composite).array;
return iota(2, limit).filter!(i => !composite[i]).array;
}
void main() {
50.sieve.writeln;
}
```
```txt
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47]
```
### Faster Version
This version uses an array of bits (instead of booleans, that are represented with one byte), and skips even numbers. The output is the same.
```d
import std.stdio, std.math, std.array;
size_t[] sieve(in size_t m) pure nothrow @safe {
if (m < 3)
return null;
immutable size_t n = m - 1;
enum size_t bpc = size_t.sizeof * 8;
auto F = new size_t[((n + 2) / 2) / bpc + 1];
F[] = size_t.max;
size_t isSet(in size_t i) nothrow @safe @nogc {
immutable size_t offset = i / bpc;
immutable size_t mask = 1 << (i % bpc);
return F[offset] & mask;
}
void resetBit(in size_t i) nothrow @safe @nogc {
immutable size_t offset = i / bpc;
immutable size_t mask = 1 << (i % bpc);
if ((F[offset] & mask) != 0)
F[offset] = F[offset] ^ mask;
}
for (size_t i = 3; i <= sqrt(real(n)); i += 2)
if (isSet((i - 3) / 2))
for (size_t j = i * i; j <= n; j += 2 * i)
resetBit((j - 3) / 2);
Appender!(size_t[]) result;
result ~= 2;
for (size_t i = 3; i <= n; i += 2)
if (isSet((i - 3) / 2))
result ~= i;
return result.data;
}
void main() {
50.sieve.writeln;
}
```
### Extensible Version
(This version is used in the task [[Extensible prime generator#D|Extensible prime generator]].)
```d
/// Extensible Sieve of Eratosthenes.
struct Prime {
uint[] a = [2];
private void grow() pure nothrow @safe {
immutable p0 = a[$ - 1] + 1;
auto b = new bool[p0];
foreach (immutable di; a) {
immutable uint i0 = p0 / di * di;
uint i = (i0 < p0) ? i0 + di - p0 : i0 - p0;
for (; i < b.length; i += di)
b[i] = true;
}
foreach (immutable uint i, immutable bi; b)
if (!b[i])
a ~= p0 + i;
}
uint opCall(in uint n) pure nothrow @safe {
while (n >= a.length)
grow;
return a[n];
}
}
version (sieve_of_eratosthenes3_main) {
void main() {
import std.stdio, std.range, std.algorithm;
Prime prime;
uint.max.iota.map!prime.until!q{a > 50}.writeln;
}
}
```
To see the output (that is the same), compile with -version=sieve_of_eratosthenes3_main.
## Dart
```dart
// helper function to pretty print an Iterable
String iterableToString(Iterable seq) {
String str = "[";
Iterator i = seq.iterator;
if (i.moveNext()) str += i.current.toString();
while(i.moveNext()) {
str += ", " + i.current.toString();
}
return str + "]";
}
main() {
int limit = 1000;
int strt = new DateTime.now().millisecondsSinceEpoch;
Set sieve = new Set();
for(int i = 2; i <= limit; i++) {
sieve.add(i);
}
for(int i = 2; i * i <= limit; i++) {
if(sieve.contains(i)) {
for(int j = i * i; j <= limit; j += i) {
sieve.remove(j);
}
}
}
var sortedValues = new List.from(sieve);
int elpsd = new DateTime.now().millisecondsSinceEpoch - strt;
print("Found " + sieve.length.toString() + " primes up to " + limit.toString() +
" in " + elpsd.toString() + " milliseconds.");
print(iterableToString(sortedValues)); // expect sieve.length to be 168 up to 1000...
// Expect.equals(168, sieve.length);
}
```
```txt
Found 168 primes up to 1000 in 9 milliseconds.
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997]
```
Although it has the characteristics of a true Sieve of Eratosthenes, the above code isn't very efficient due to the remove/modify operations on the Set. Due to these, the computational complexity isn't close to linear with increasing range and it is quite slow for larger sieve ranges compared to compiled languages, taking an average of about 22 thousand CPU clock cycles for each of the 664579 primes (about 4 seconds on a 3.6 Gigahertz CPU) just to sieve to ten million.
===faster bit-packed array odds-only solution===
```dart
import 'dart:typed_data';
import 'dart:math';
Iterable soeOdds(int limit) {
if (limit < 3) return limit < 2 ? Iterable.empty() : [2];
int lmti = (limit - 3) >> 1;
int bfsz = (lmti >> 3) + 1;
int sqrtlmt = (sqrt(limit) - 3).floor() >> 1;
Uint32List cmpsts = Uint32List(bfsz);
for (int i = 0; i <= sqrtlmt; ++i)
if ((cmpsts[i >> 5] & (1 << (i & 31))) == 0) {
int p = i + i + 3;
for (int j = (p * p - 3) >> 1; j <= lmti; j += p)
cmpsts[j >> 5] |= 1 << (j & 31);
}
return
[2].followedBy(
Iterable.generate(lmti + 1)
.where((i) => cmpsts[i >> 5] & (1 << (i & 31)) == 0)
.map((i) => i + i + 3) );
}
void main() {
final int range = 100000000;
String s = "( ";
primesPaged().take(25).forEach((p)=>s += "$p "); print(s + ")");
print("There are ${countPrimesTo(1000000)} primes to 1000000.");
final start = DateTime.now().millisecondsSinceEpoch;
final answer = soeOdds(range).length;
final elapsed = DateTime.now().millisecondsSinceEpoch - start;
print("There were $answer primes found up to $range.");
print("This test bench took $elapsed milliseconds.");
}
```
```txt
( 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 )
There are 78498 primes to 1000000.
There were 5761455 primes found up to 100000000.
This test bench took 4604 milliseconds.
```
The above code is somewhat faster at about 1.5 thousand CPU cycles per prime here run on a 1.92 Gigahertz low end Intel x5-Z8350 CPU or about 2.5 seconds on a 3.6 Gigahertz CPU using the Dart VM to sieve to 100 million.
### Unbounded infinite iterators/generators of primes
'''Infinite generator using a (hash) Map (sieves odds-only)'''
The following code will have about O(n log (log n)) performance due to a hash table having O(1) average performance and is only somewhat slow due to the constant overhead of processing hashes:
```dart>Iterable oddprms() sync* {
yield(3); yield(5); // need at least 2 for initialization
final Map bpmap = {9: 6};
final Iterator bps = oddprms().iterator;
bps.moveNext(); bps.moveNext(); // skip past 3 to 5
int bp = bps.current;
int n = bp;
int q = bp * bp;
while (true) {
n += 2;
while (n >= q || bpmap.containsKey(n)) {
if (n >= q) {
final int inc = bp << 1;
bpmap[bp * bp + inc] = inc;
bps.moveNext(); bp = bps.current; q = bp * bp;
} else {
final int inc = bpmap.remove(n);
int next = n + inc;
while (bpmap.containsKey(next)) {
next += inc;
}
bpmap[next] = inc;
}
n += 2;
}
yield(n);
}
}
return [2].followedBy(oddprms());
}
void main() {
final int range = 100000000;
String s = "( ";
primesMap().take(25).forEach((p)=>s += "$p "); print(s + ")");
print("There are ${primesMap().takeWhile((p)=>p<=1000000).length} preimes to 1000000.");
final start = DateTime.now().millisecondsSinceEpoch;
final answer = primesMap().takeWhile((p)=>p<=range).length;
final elapsed = DateTime.now().millisecondsSinceEpoch - start;
print("There were $answer primes found up to $range.");
print("This test bench took $elapsed milliseconds.");
}
```
```txt
( 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 )
There are 78498 preimes to 1000000.
There were 5761455 primes found up to 100000000.
This test bench took 16086 milliseconds.
```
This takes about 5300 CPU clock cycles per prime or about 8.4 seconds if run on a 3.6 Gigahertz CPU, which is slower than the above fixed bit-packed array version but has the advantage that it runs indefinitely, (at least on 64-bit machines; on 32 bit machines it can only be run up to the 32-bit number range, or just about a factor of 20 above as above).
Due to the constant execution overhead this is only reasonably useful for ranges up to tens of millions anyway.
'''Fast page segmented array infinite generator (sieves odds-only)'''
The following code also theoretically has a O(n log (log n)) execution speed performance and the same limited use on 32-bit execution platformas, but won't realize the theoretical execution complexity for larger primes due to the cache size increasing in size beyond its limits; but as the CPU L2 cache size that it automatically grows to use isn't any slower than the basic culling loop speed, it won't slow down much above that limit up to ranges of about 2.56e14, which will take in the order of weeks:
```dart
import 'dart:typed_data';
import 'dart:math';
import 'dart:collection';
// a lazy list
typedef _LazyList _Thunk();
class _LazyList {
final T head;
_Thunk thunk;
_LazyList _rest;
_LazyList(T this.head, _Thunk this.thunk);
_LazyList get rest {
if (this.thunk != null) {
this._rest = this.thunk();
this.thunk = null;
}
return this._rest;
}
}
class _LazyListIterable extends IterableBase {
_LazyList _first;
_LazyListIterable(_LazyList this._first);
@override Iterator get iterator {
Iterable inner() sync* {
_LazyList current = this._first;
while (true) {
yield(current.head);
current = current.rest;
}
}
return inner().iterator;
}
}
// zero bit population count Look Up Table for 16-bit range...
final Uint8List CLUT =
Uint8List.fromList(
Iterable.generate(65536)
.map((i) {
final int v0 = ~i & 0xFFFF;
final int v1 = v0 - ((v0 & 0xAAAA) >> 1);
final int v2 = (v1 & 0x3333) + ((v1 & 0xCCCC) >> 2);
return (((((v2 & 0x0F0F) + ((v2 & 0xF0F0) >> 4)) * 0x0101)) >> 8) & 31;
})
.toList());
int _countComposites(Uint8List cmpsts) {
Uint16List buf = Uint16List.view(cmpsts.buffer);
int lmt = buf.length;
int count = 0;
for (var i = 0; i < lmt; ++i) {
count += CLUT[buf[i]];
}
return count;
}
// converts an entire sieved array of bytes into an array of UInt32 primes,
// to be used as a source of base primes...
Uint32List _composites2BasePrimeArray(int low, Uint8List cmpsts) {
final int lmti = cmpsts.length << 3;
final int len = _countComposites(cmpsts);
final Uint32List rslt = Uint32List(len);
int j = 0;
for (int i = 0; i < lmti; ++i) {
if (cmpsts[i >> 3] & 1 << (i & 7) == 0) {
rslt[j++] = low + i + i;
}
}
return rslt;
}
// do sieving work based on low starting value for the given buffer and
// the given lazy list of base prime arrays...
void _sieveComposites(int low, Uint8List buffer, Iterable bpas) {
final int lowi = (low - 3) >> 1;
final int len = buffer.length;
final int lmti = len << 3;
final int nxti = lowi + lmti;
for (var bpa in bpas) {
for (var bp in bpa) {
final int bpi = (bp - 3) >> 1;
int strti = ((bpi * (bpi + 3)) << 1) + 3;
if (strti >= nxti) return;
if (strti >= lowi) strti = strti - lowi;
else {
strti = (lowi - strti) % bp;
if (strti != 0) strti = bp - strti;
}
if (bp <= len >> 3 && strti <= lmti - bp << 6) {
final int slmti = min(lmti, strti + bp << 3);
for (var s = strti; s < slmti; s += bp) {
final int msk = 1 << (s & 7);
for (var c = s >> 3; c < len; c += bp) {
buffer[c] |= msk;
}
}
}
else {
for (var c = strti; c < lmti; c += bp) {
buffer[c >> 3] |= 1 << (c & 7);
}
}
}
}
}
// starts the secondary base primes feed with minimum size in bits set to 4K...
// thus, for the first buffer primes up to 8293,
// the seeded primes easily cover it as 97 squared is 9409...
Iterable _makeBasePrimeArrays() {
var cmpsts = Uint8List(512);
_LazyList _nextelem(int low, Iterable bpas) {
// calculate size so that the bit span is at least as big as the
// maximum culling prime required, rounded up to minsizebits blocks...
final int rqdsz = 2 + sqrt((1 + low).toDouble()).toInt();
final sz = (((rqdsz >> 12) + 1) << 9); // size in bytes
if (sz > cmpsts.length) cmpsts = Uint8List(sz);
cmpsts.fillRange(0, cmpsts.length, 0);
_sieveComposites(low, cmpsts, bpas);
final arr = _composites2BasePrimeArray(low, cmpsts);
final nxt = low + (cmpsts.length << 4);
return _LazyList(arr, () => _nextelem(nxt, bpas));
}
// pre-seeding breaks recursive race,
// as only known base primes used for first page...
final preseedarr = Uint32List.fromList( [ // pre-seed to 100, can sieve to 10,000...
3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41
, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 ] );
return _LazyListIterable(
_LazyList(preseedarr,
() => _nextelem(101, _makeBasePrimeArrays()))
);
}
// an iterable sequence over successive sieved buffer composite arrays,
// returning a tuple of the value represented by the lowest possible prime
// in the sieved composites array and the array itself;
// the array has a 16 Kilobytes minimum size (CPU L1 cache), but
// will grow so that the bit span is larger than the
// maximum culling base prime required, possibly making it larger than
// the L1 cache for large ranges, but still reasonably efficient using
// the L2 cache: very efficient up to about 16e9 range;
// reasonably efficient to about 2.56e14 for two Megabyte L2 cache = > 1 day...
Iterable _makeSievePages() sync* {
final bpas = _makeBasePrimeArrays(); // secondary source of base prime arrays
int low = 3;
Uint8List cmpsts = Uint8List(16384);
_sieveComposites(3, cmpsts, bpas);
while (true) {
yield([low, cmpsts]);
final rqdsz = 2 + sqrt((1 + low).toDouble()).toInt(); // problem with sqrt not exact past about 10^12!!!!!!!!!
final sz = ((rqdsz >> 17) + 1) << 14; // size iin bytes
if (sz > cmpsts.length) cmpsts = Uint8List(sz);
cmpsts.fillRange(0, cmpsts.length, 0);
low += cmpsts.length << 4;
_sieveComposites(low, cmpsts, bpas);
}
}
int countPrimesTo(int range) {
if (range < 3) { if (range < 2) return 0; else return 1; }
var count = 1;
for (var sp in _makeSievePages()) {
int low = sp[0]; Uint8List cmpsts = sp[1];
if ((low + (cmpsts.length << 4)) > range) {
int lsti = (range - low) >> 1;
var lstw = (lsti >> 4); var lstb = lstw << 1;
var msk = (-2 << (lsti & 15)) & 0xFFFF;
var buf = Uint16List.view(cmpsts.buffer, 0, lstw);
for (var i = 0; i < lstw; ++i)
count += CLUT[buf[i]];
count += CLUT[(cmpsts[lstb + 1] << 8) | cmpsts[lstb] | msk];
break;
} else {
count += _countComposites(cmpsts);
}
}
return count;
}
// sequence over primes from above page iterator;
// unless doing something special with individual primes, usually unnecessary;
// better to do manipulations based on the composites bit arrays...
// takes at least as long to enumerate the primes as sieve them...
Iterable primesPaged() sync* {
yield(2);
for (var sp in _makeSievePages()) {
int low = sp[0]; Uint8List cmpsts = sp[1];
var szbts = cmpsts.length << 3;
for (var i = 0; i < szbts; ++i) {
if (cmpsts[i >> 3].toInt() & (1 << (i & 7)) != 0) continue;
yield(low + i + i);
}
}
}
void main() {
final int range = 1000000000;
String s = "( ";
primesPaged().take(25).forEach((p)=>s += "$p "); print(s + ")");
print("There are ${countPrimesTo(1000000)} primes to 1000000.");
final start = DateTime.now().millisecondsSinceEpoch;
final answer = countPrimesTo(range); // fast way
// final answer = primesPaged().takeWhile((p)=>p<=range).length; // slow way using enumeration
final elapsed = DateTime.now().millisecondsSinceEpoch - start;
print("There were $answer primes found up to $range.");
print("This test bench took $elapsed milliseconds.");
}
```
```txt
( 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 )
There are 78498 primes to 1000000.
There were 50847534 primes found up to 1000000000.
This test bench took 9385 milliseconds.
```
This version counts the primes up to one billion in about five seconds at 3.6 Gigahertz (a low end 1.92 Gigahertz CPU used here) or about 350 CPU clock cycles per prime under the Dart Virtual Machine (VM).
Note that it takes about four times as long to do this using the provided primes generator/enumerator as noted in the code, which is normal for all languages that it takes longer to actually enumerate the primes than it does to sieve in culling the composite numbers, but Dart is somewhat slower than most for this.
The algorithm can be sped up by a factor of four by extreme wheel factorization and (likely) about a factor of the effective number of CPU cores by using multi-processing isolates, but there isn't much point if one is to use the prime generator for output. For most purposes, it is better to use custom functions that directly manipulate the culled bit-packed page segments as `countPrimesTo` does here.
## Delphi
```delphi
program erathostenes;
{$APPTYPE CONSOLE}
type
TSieve = class
private
fPrimes: TArray;
procedure InitArray;
procedure Sieve;
function getNextPrime(aStart: integer): integer;
function getPrimeArray: TArray;
public
function getPrimes(aMax: integer): TArray;
end;
{ TSieve }
function TSieve.getNextPrime(aStart: integer): integer;
begin
result := aStart;
while not fPrimes[result] do
inc(result);
end;
function TSieve.getPrimeArray: TArray;
var
i, n: integer;
begin
n := 0;
setlength(result, length(fPrimes)); // init array with maximum elements
for i := 2 to high(fPrimes) do
begin
if fPrimes[i] then
begin
result[n] := i;
inc(n);
end;
end;
setlength(result, n); // reduce array to actual elements
end;
function TSieve.getPrimes(aMax: integer): TArray;
begin
setlength(fPrimes, aMax);
InitArray;
Sieve;
result := getPrimeArray;
end;
procedure TSieve.InitArray;
begin
for i := 2 to high(fPrimes) do
fPrimes[i] := true;
end;
procedure TSieve.Sieve;
var
i, n, max: integer;
begin
max := length(fPrimes);
i := 2;
while i < sqrt(max) do
begin
n := sqr(i);
while n < max do
begin
fPrimes[n] := false;
inc(n, i);
end;
i := getNextPrime(i + 1);
end;
end;
var
i: integer;
Sieve: TSieve;
begin
Sieve := TSieve.Create;
for i in Sieve.getPrimes(100) do
write(i, ' ');
Sieve.Free;
readln;
end.
```
Output:
```txt
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
```
## DWScript
```delphi
function Primes(limit : Integer) : array of Integer;
var
n, k : Integer;
sieve := new Boolean[limit+1];
begin
for n := 2 to Round(Sqrt(limit)) do begin
if not sieve[n] then begin
for k := n*n to limit step n do
sieve[k] := True;
end;
end;
for k:=2 to limit do
if not sieve[k] then
Result.Add(k);
end;
var r := Primes(50);
var i : Integer;
for i:=0 to r.High do
PrintLn(r[i]);
```
## Dylan
With outer to sqrt and inner to p^2 optimizations:
```dylan
define method primes(n)
let limit = floor(n ^ 0.5) + 1;
let sieve = make(limited(, of: ), size: n + 1, fill: #t);
let last-prime = 2;
while (last-prime < limit)
for (x from last-prime ^ 2 to n by last-prime)
sieve[x] := #f;
end for;
block (found-prime)
for (n from last-prime + 1 below limit)
if (sieve[n] = #f)
last-prime := n;
found-prime()
end;
end;
last-prime := limit;
end block;
end while;
for (x from 2 to n)
if (sieve[x]) format-out("Prime: %d\n", x); end;
end;
end;
```
## E
E's standard library doesn't have a step-by-N numeric range, so we'll define one, implementing the standard iteration protocol.
def rangeFromBelowBy(start, limit, step) {
return def stepper {
to iterate(f) {
var i := start
while (i < limit) {
f(null, i)
i += step
}
}
}
}
The sieve itself:
def eratosthenes(limit :(int > 2), output) {
def composite := [].asSet().diverge()
for i ? (!composite.contains(i)) in 2..!limit {
output(i)
composite.addAll(rangeFromBelowBy(i ** 2, limit, i))
}
}
Example usage:
? eratosthenes(12, println)
# stdout: 2
# 3
# 5
# 7
# 11
## EasyLang
len prims[] 100
max = floor sqrt len prims[]
tst = 2
while tst <= max
if prims[tst] = 0
i = tst * tst
while i < len prims[]
prims[i] = 1
i += tst
.
.
tst += 1
.
i = 2
while i < len prims[]
if prims[i] = 0
print i
.
i += 1
.
```
## eC
Note: this is not a Sieve of Eratosthenes; it is just trial division.
```cpp
public class FindPrime
{
Array primeList { [ 2 ], minAllocSize = 64 };
int index;
index = 3;
bool HasPrimeFactor(int x)
{
int max = (int)floor(sqrt((double)x));
for(i : primeList)
{
if(i > max) break;
if(x % i == 0) return true;
}
return false;
}
public int GetPrime(int x)
{
if(x > primeList.count - 1)
{
for (; primeList.count != x; index += 2)
if(!HasPrimeFactor(index))
{
if(primeList.count >= primeList.minAllocSize) primeList.minAllocSize *= 2;
primeList.Add(index);
}
}
return primeList[x-1];
}
}
class PrimeApp : Application
{
FindPrime fp { };
void Main()
{
int num = argc > 1 ? atoi(argv[1]) : 1;
PrintLn(fp.GetPrime(num));
}
}
```
## EchoLisp
### Sieve
```lisp
(require 'types) ;; bit-vector
;; converts sieve->list for integers in [nmin .. nmax[
(define (s-range sieve nmin nmax (base 0))
(for/list ([ i (in-range nmin nmax)]) #:when (bit-vector-ref sieve i) (+ i base)))
;; next prime in sieve > p, or #f
(define (s-next-prime sieve p ) ;;
(bit-vector-scan-1 sieve (1+ p)))
;; returns a bit-vector - sieve- all numbers in [0..n[
(define (eratosthenes n)
(define primes (make-bit-vector-1 n ))
(bit-vector-set! primes 0 #f)
(bit-vector-set! primes 1 #f)
(for ([p (1+ (sqrt n))])
#:when (bit-vector-ref primes p)
(for ([j (in-range (* p p) n p)])
(bit-vector-set! primes j #f)))
primes)
(define s-primes (eratosthenes 10_000_000))
(s-range s-primes 0 100)
→ (2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97)
(s-range s-primes 1_000_000 1_000_100)
→ (1000003 1000033 1000037 1000039 1000081 1000099)
(s-next-prime s-primes 9_000_000)
→ 9000011
```
### Segmented sieve
Allow to extend the basis sieve (n) up to n^2. Memory requirement is O(√n)
```scheme
;; ref : http://research.cs.wisc.edu/techreports/1990/TR909.pdf
;; delta multiple of sqrt(n)
;; segment is [left .. left+delta-1]
(define (segmented sieve left delta (p 2) (first 0))
(define segment (make-bit-vector-1 delta))
(define right (+ left (1- delta)))
(define pmax (sqrt right))
(while p
#:break (> p pmax)
(set! first (+ left (modulo (- p (modulo left p)) p )))
(for [(q (in-range first (1+ right) p))]
(bit-vector-set! segment (- q left) #f))
(set! p (bit-vector-scan-1 sieve (1+ p))))
segment)
(define (seg-range nmin delta)
(s-range (segmented s-primes nmin delta) 0 delta nmin))
(seg-range 10_000_000_000 1000) ;; 15 milli-sec
→ (10000000019 10000000033 10000000061 10000000069 10000000097 10000000103 10000000121
10000000141 10000000147 10000000207 10000000259 10000000277 10000000279 10000000319
10000000343 10000000391 10000000403 10000000469 10000000501 10000000537 10000000583
10000000589 10000000597 10000000601 10000000631 10000000643 10000000649 10000000667
10000000679 10000000711 10000000723 10000000741 10000000753 10000000793 10000000799
10000000807 10000000877 10000000883 10000000889 10000000949 10000000963 10000000991
10000000993 10000000999)
;; 8 msec using the native (prime?) function
(for/list ((p (in-range 1_000_000_000 1_000_001_000))) #:when (prime? p) p)
```
### Wheel
A 2x3 wheel gives a 50% performance gain.
```scheme
;; 2x3 wheel
(define (weratosthenes n)
(define primes (make-bit-vector n )) ;; everybody to #f (false)
(bit-vector-set! primes 2 #t)
(bit-vector-set! primes 3 #t)
(bit-vector-set! primes 5 #t)
(for ([i (in-range 6 n 6) ]) ;; set candidate primes
(bit-vector-set! primes (1+ i) #t)
(bit-vector-set! primes (+ i 5) #t)
)
(for ([p (in-range 5 (1+ (sqrt n)) 2 ) ])
#:when (bit-vector-ref primes p)
(for ([j (in-range (* p p) n p)])
(bit-vector-set! primes j #f)))
primes)
```
## Eiffel
```eiffel
class
APPLICATION
create
make
feature
make
-- Run application.
do
across primes_through (100) as ic loop print (ic.item.out + " ") end
end
primes_through (a_limit: INTEGER): LINKED_LIST [INTEGER]
-- Prime numbers through `a_limit'
require
valid_upper_limit: a_limit >= 2
local
l_tab: ARRAY [BOOLEAN]
do
create Result.make
create l_tab.make_filled (True, 2, a_limit)
across
l_tab as ic
loop
if ic.item then
Result.extend (ic.target_index)
across ((ic.target_index * ic.target_index) |..| l_tab.upper).new_cursor.with_step (ic.target_index) as id
loop
l_tab [id.item] := False
end
end
end
end
end
```
Output:
```txt
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
```
## Elixir
```elixir
defmodule Prime do
def eratosthenes(limit \\ 1000) do
sieve = [false, false | Enum.to_list(2..limit)] |> List.to_tuple
check_list = [2 | Stream.iterate(3, &(&1+2)) |> Enum.take(round(:math.sqrt(limit)/2))]
Enum.reduce(check_list, sieve, fn i,tuple ->
if elem(tuple,i) do
clear_num = Stream.iterate(i*i, &(&1+i)) |> Enum.take_while(fn x -> x <= limit end)
clear(tuple, clear_num)
else
tuple
end
end)
end
defp clear(sieve, list) do
Enum.reduce(list, sieve, fn i, acc -> put_elem(acc, i, false) end)
end
end
limit = 199
sieve = Prime.eratosthenes(limit)
Enum.each(0..limit, fn n ->
if x=elem(sieve, n), do: :io.format("~3w", [x]), else: :io.format(" .")
if rem(n+1, 20)==0, do: IO.puts ""
end)
```
```txt
. . 2 3 . 5 . 7 . . . 11 . 13 . . . 17 . 19
. . . 23 . . . . . 29 . 31 . . . . . 37 . .
. 41 . 43 . . . 47 . . . . . 53 . . . . . 59
. 61 . . . . . 67 . . . 71 . 73 . . . . . 79
. . . 83 . . . . . 89 . . . . . . . 97 . .
.101 .103 . . .107 .109 . . .113 . . . . . .
. . . . . . .127 . . .131 . . . . .137 .139
. . . . . . . . .149 .151 . . . . .157 . .
. . .163 . . .167 . . . . .173 . . . . .179
.181 . . . . . . . . .191 .193 . . .197 .199
```
Shorter version (but slow):
```elixir
defmodule Sieve do
def primes_to(limit), do: sieve(Enum.to_list(2..limit))
defp sieve([h|t]), do: [h|sieve(t -- for n <- 1..length(t), do: h*n)]
defp sieve([]), do: []
end
```
'''Alternate much faster odds-only version more suitable for immutable data structures using a (hash) Map'''
The above code has a very limited useful range due to being very slow: for example, to sieve to a million, even changing the algorithm to odds-only, requires over 800 thousand "copy-on-update" operations of the entire saved immutable tuple ("array") of 500 thousand bytes in size, making it very much a "toy" application. The following code overcomes that problem by using a (immutable/hashed) Map to store the record of the current state of the composite number chains resulting from each of the secondary streams of base primes, which are only 167 in number up to this range; it is a functional "incremental" Sieve of Eratosthenes implementation:
```elixir
defmodule PrimesSoEMap do
@typep stt :: {integer, integer, integer, Enumerable.integer, %{integer => integer}}
@spec advance(stt) :: stt
defp advance {n, bp, q, bps?, map} do
bps = if bps? === nil do Stream.drop(oddprms(), 1) else bps? end
nn = n + 2
if nn >= q do
inc = bp + bp
nbps = bps |> Stream.drop(1)
[nbp] = nbps |> Enum.take(1)
advance {nn, nbp, nbp * nbp, nbps, map |> Map.put(nn + inc, inc)}
else if Map.has_key?(map, nn) do
{inc, rmap} = Map.pop(map, nn)
[next] =
Stream.iterate(nn + inc, &(&1 + inc))
|> Stream.drop_while(&(Map.has_key?(rmap, &1))) |> Enum.take(1)
advance {nn, bp, q, bps, Map.put(rmap, next, inc)}
else
{nn, bp, q, bps, map}
end end
end
@spec oddprms() :: Enumerable.integer
defp oddprms do # put first base prime cull seq in Map so never empty
# advance base odd primes to 5 when initialized
init = {7, 5, 25, nil, %{9 => 6}}
[3, 5] # to avoid race, preseed with the first 2 elements...
|> Stream.concat(
Stream.iterate(init, &(advance &1))
|> Stream.map(fn {p,_,_,_,_} -> p end))
end
@spec primes() :: Enumerable.integer
def primes do
Stream.concat([2], oddprms())
end
end
range = 1000000
IO.write "The first 25 primes are:\n( "
PrimesSoEMap.primes() |> Stream.take(25) |> Enum.each(&(IO.write "#{&1} "))
IO.puts ")"
testfunc =
fn () ->
ans =
PrimesSoEMap.primes() |> Stream.take_while(&(&1 <= range)) |> Enum.count()
ans end
:timer.tc(testfunc)
|> (fn {t,ans} ->
IO.puts "There are #{ans} primes up to #{range}."
IO.puts "This test bench took #{t} microseconds." end).()
```
```txt
The first 25 primes are:
( 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 )
There are 78498 primes up to 1000000.
This test bench took 3811957 microseconds.
```
The output time of about 3.81 seconds to one million is on a 1.92 Gigahertz CPU meaning that it takes about 93 thousand CPU clock cycles per prime which is still quite slow compared to mutable data structure implementations but comparable to "functional" implementations in other languages and is slow due to the time to calculate the required hashes. One advantage that it has is that it is O(n log (log n)) asymptotic computational complexity meaning that it takes not much more than ten times as long to sieve a range ten times higher.
This algorithm could be easily changed to use a Priority Queue (preferably Min-Heap based for the least constant factor computational overhead) to save some of the computation time, but then it will have the same computational complexity as the following code and likely about the same execution time.
'''Alternate faster odds-only version more suitable for immutable data structures using lazy Streams of Co-Inductive Streams'''
In order to save the computation time of computing the hashes, the following version uses a deferred execution Co-Inductive Stream type (constructed using Tuple's) in an infinite tree folding structure (by the `pairs` function):
```elixir
defmodule PrimesSoETreeFolding do
@typep cis :: {integer, (() -> cis)}
@typep ciss :: {cis, (() -> ciss)}
@spec merge(cis, cis) :: cis
defp merge(xs, ys) do
{x, restxs} = xs; {y, restys} = ys
cond do
x < y -> {x, fn () -> merge(restxs.(), ys) end}
y < x -> {y, fn () -> merge(xs, restys.()) end}
true -> {x, fn () -> merge(restxs.(), restys.()) end}
end
end
@spec smlt(integer, integer) :: cis
defp smlt(c, inc) do
{c, fn () -> smlt(c + inc, inc) end}
end
@spec smult(integer) :: cis
defp smult(p) do
smlt(p * p, p + p)
end
P
@spec allmults(cis) :: ciss
defp allmults {p, restps} do
{smult(p), fn () -> allmults(restps.()) end}
end
@spec pairs(ciss) :: ciss
defp pairs {cs0, restcss0} do
{cs1, restcss1} = restcss0.()
{merge(cs0, cs1), fn () -> pairs(restcss1.()) end}
end
@spec cmpsts(ciss) :: cis
defp cmpsts {cs, restcss} do
{c, restcs} = cs
{c, fn () -> merge(restcs.(), cmpsts(pairs(restcss.()))) end}
end
@spec minusat(integer, cis) :: cis
defp minusat(n, cmps) do
{c, restcs} = cmps
if n < c do
{n, fn () -> minusat(n + 2, cmps) end}
else
minusat(n + 2, restcs.())
end
end
@spec oddprms() :: cis
defp oddprms() do
{3, fn () ->
{5, fn () -> minusat(7, cmpsts(allmults(oddprms()))) end}
end}
end
@spec primes() :: Enumerable.t
def primes do
[2] |> Stream.concat(
Stream.iterate(oddprms(), fn {_, restps} -> restps.() end)
|> Stream.map(fn {p, _} -> p end)
)
end
end
range = 1000000
IO.write "The first 25 primes are:\n( "
PrimesSoETreeFolding.primes() |> Stream.take(25) |> Enum.each(&(IO.write "#{&1} "))
IO.puts ")"
testfunc =
fn () ->
ans =
PrimesSoETreeFolding.primes() |> Stream.take_while(&(&1 <= range)) |> Enum.count()
ans end
:timer.tc(testfunc)
|> (fn {t,ans} ->
IO.puts "There are #{ans} primes up to #{range}."
IO.puts "This test bench took #{t} microseconds." end).()
```
It's output is identical to the previous version other than the time required is less than half; however, it has a O(n (log n) (log (log n))) asymptotic computation complexity meaning that it gets slower with range faster than the above version. That said, it would take sieving to billions taking hours before the two would take about the same time.
## Emacs Lisp
```lisp
(defun sieve-set (limit)
(let ((xs (make-vector (1+ limit) 0)))
(loop for i from 2 to limit
when (zerop (aref xs i))
collect i
and do (loop for m from (* i i) to limit by i
do (aset xs m 1)))))
```
Straightforward implementation of [http://en.wikipedia.org/wiki/Sieve_of_Eratosthenes#Implementation sieve of Eratosthenes], 2 times faster:
```lisp
(defun sieve (limit)
(let ((xs (vconcat [0 0] (number-sequence 2 limit))))
(loop for i from 2 to (sqrt limit)
when (aref xs i)
do (loop for m from (* i i) to limit by i
do (aset xs m 0)))
(remove 0 xs)))
```
## Erlang
### Erlang using Dicts
```Erlang
-module( sieve_of_eratosthenes ).
-export( [primes_upto/1] ).
primes_upto( N ) ->
Ns = lists:seq( 2, N ),
Dict = dict:from_list( [{X, potential_prime} || X <- Ns] ),
{Upto_sqrt_ns, _T} = lists:split( erlang:round(math:sqrt(N)), Ns ),
{N, Prime_dict} = lists:foldl( fun find_prime/2, {N, Dict}, Upto_sqrt_ns ),
lists:sort( dict:fetch_keys(Prime_dict) ).
find_prime( N, {Max, Dict} ) -> find_prime( dict:find(N, Dict), N, {Max, Dict} ).
find_prime( error, _N, Acc ) -> Acc;
find_prime( {ok, _Value}, N, {Max, Dict} ) -> {Max, lists:foldl( fun dict:erase/2, Dict, lists:seq(N*N, Max, N) )}.
```
```txt
35> sieve_of_eratosthenes:primes_upto( 20 ).
[2,3,5,7,11,13,17,19]
```
===Erlang Lists of Tuples, Sloww===
A much slower, perverse method, using only lists of tuples. Especially evil is the P = lists:filtermap operation which yields a list for every iteration of the X * M row.
Has the virtue of working for any -> N :)
```Erlang
-module( sieve ).
-export( [main/1,primes/2] ).
main(N) -> io:format("Primes: ~w~n", [ primes(2,N) ]).
primes(M,N) -> primes(M, N,lists:seq( M, N ),[]).
primes(M,N,_Acc,Tuples) when M > N/2-> out(Tuples);
primes(M,N,Acc,Tuples) when length(Tuples) < 1 ->
primes(M,N,Acc,[{X, X} || X <- Acc]);
primes(M,N,Acc,Tuples) ->
{SqrtN, _T} = lists:split( erlang:round(math:sqrt(N)), Acc ),
F = Tuples,
Ms = lists:filtermap(fun(X) -> if X > 0 -> {true, X * M}; true -> false end end, SqrtN),
P = lists:filtermap(fun(T) ->
case lists:keymember(T,1,F) of true ->
{true, lists:keyreplace(T,1,F,{T,0})};
_-> false end end, Ms),
AA = mergeT(P,lists:last(P),1 ),
primes(M+1,N,Acc,AA).
mergeT(L,M,Acc) when Acc == length(L) -> M;
mergeT(L,M,Acc) ->
A = lists:nth(Acc,L),
B = M,
Mer = lists:zipwith(fun(X, Y) -> if X < Y -> X; true -> Y end end, A, B),
mergeT(L,Mer,Acc+1).
out(Tuples) ->
Primes = lists:filter( fun({_,Y}) -> Y > 0 end, Tuples),
[ X || {X,_} <- Primes ].
```
```txt
109> sieve:main(20).
Primes: [2,3,5,7,11,13,17,19]
ok
110> timer:tc(sieve, main, [20]).
Primes: [2,3,5,7,11,13,17,19]
{129,ok}
```
### Erlang with ordered sets
Since I had written a really odd and slow one, I thought I'd best do a better performer. Inspired by an example from https://github.com/jupp0r
```Erlang
-module(ossieve).
-export([main/1]).
sieve(Candidates,SearchList,Primes,_Maximum) when length(SearchList) == 0 ->
ordsets:union(Primes,Candidates);
sieve(Candidates,SearchList,Primes,Maximum) ->
H = lists:nth(1,string:substr(Candidates,1,1)),
Reduced1 = ordsets:del_element(H, Candidates),
{Reduced2, ReducedSearch} = remove_multiples_of(H, Reduced1, SearchList),
NewPrimes = ordsets:add_element(H,Primes),
sieve(Reduced2, ReducedSearch, NewPrimes, Maximum).
remove_multiples_of(Number,Candidates,SearchList) ->
NewSearchList = ordsets:filter( fun(X) -> X >= Number * Number end, SearchList),
RemoveList = ordsets:filter( fun(X) -> X rem Number == 0 end, NewSearchList),
{ordsets:subtract(Candidates, RemoveList), ordsets:subtract(NewSearchList, RemoveList)}.
main(N) ->
io:fwrite("Creating Candidates...~n"),
CandidateList = lists:seq(3,N,2),
Candidates = ordsets:from_list(CandidateList),
io:fwrite("Sieving...~n"),
ResultSet = ordsets:add_element(2,sieve(Candidates,Candidates,ordsets:new(),N)),
io:fwrite("Sieved... ~w~n",[ResultSet]).
```
```txt
36> ossieve:main(100).
Creating Candidates...
Sieving...
Sieved... [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97]
ok
```
### Erlang Canonical
A pure list comprehension approach.
```Erlang
-module(sieveof).
-export([main/1,primes/1, primes/2]).
main(X) -> io:format("Primes: ~w~n", [ primes(X) ]).
primes(X) -> sieve(range(2, X)).
primes(X, Y) -> remove(primes(X), primes(Y)).
range(X, X) -> [X];
range(X, Y) -> [X | range(X + 1, Y)].
sieve([X]) -> [X];
sieve([H | T]) -> [H | sieve(remove([H * X || X <-[H | T]], T))].
remove(_, []) -> [];
remove([H | X], [H | Y]) -> remove(X, Y);
remove(X, [H | Y]) -> [H | remove(X, Y)].
```
{out}
```txt
> timer:tc(sieve, main, [100]).
Primes: [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97]
{7350,ok}
61> timer:tc(sieveof, main, [100]).
Primes: [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97]
{363,ok}
```
Clearly not only more elegant, but faster :) Thanks to http://stackoverflow.com/users/113644/g-b
### Erlang ets + cpu distributed implementation
much faster previous erlang examples
```Erlang
#!/usr/bin/env escript
%% -*- erlang -*-
%%! -smp enable -sname p10_4
% vim:syn=erlang
-mode(compile).
main([N0]) ->
N = list_to_integer(N0),
ets:new(comp, [public, named_table, {write_concurrency, true} ]),
ets:new(prim, [public, named_table, {write_concurrency, true}]),
composite_mc(N),
primes_mc(N),
io:format("Answer: ~p ~n", [lists:sort([X||{X,_}<-ets:tab2list(prim)])]).
primes_mc(N) ->
case erlang:system_info(schedulers) of
1 -> primes(N);
C -> launch_primes(lists:seq(1,C), C, N, N div C)
end.
launch_primes([1|T], C, N, R) -> P = self(), spawn(fun()-> primes(2,R), P ! {ok, prm} end), launch_primes(T, C, N, R);
launch_primes([H|[]], C, N, R)-> P = self(), spawn(fun()-> primes(R*(H-1)+1,N), P ! {ok, prm} end), wait_primes(C);
launch_primes([H|T], C, N, R) -> P = self(), spawn(fun()-> primes(R*(H-1)+1,R*H), P ! {ok, prm} end), launch_primes(T, C, N, R).
wait_primes(0) -> ok;
wait_primes(C) ->
receive
{ok, prm} -> wait_primes(C-1)
after 1000 -> wait_primes(C)
end.
primes(N) -> primes(2, N).
primes(I,N) when I =< N ->
case ets:lookup(comp, I) of
[] -> ets:insert(prim, {I,1})
;_ -> ok
end,
primes(I+1, N);
primes(I,N) when I > N -> ok.
composite_mc(N) -> composite_mc(N,2,round(math:sqrt(N)),erlang:system_info(schedulers)).
composite_mc(N,I,M,C) when I =< M, C > 0 ->
C1 = case ets:lookup(comp, I) of
[] -> comp_i_mc(I*I, I, N), C-1
;_ -> C
end,
composite_mc(N,I+1,M,C1);
composite_mc(_,I,M,_) when I > M -> ok;
composite_mc(N,I,M,0) ->
receive
{ok, cim} -> composite_mc(N,I,M,1)
after 1000 -> composite_mc(N,I,M,0)
end.
comp_i_mc(J, I, N) ->
Parent = self(),
spawn(fun() ->
comp_i(J, I, N),
Parent ! {ok, cim}
end).
comp_i(J, I, N) when J =< N -> ets:insert(comp, {J, 1}), comp_i(J+I, I, N);
comp_i(J, _, N) when J > N -> ok.
```
```txt
mkh@mkh-xps:~/work/mblog/pr_euler/p10$ ./generator.erl 100
Answer: [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,
97]
```
another several erlang implementation: http://mijkenator.github.io/2015/11/29/project-euler-problem-10/
## ERRE
```ERRE
PROGRAM SIEVE_ORG
! --------------------------------------------------
! Eratosthenes Sieve Prime Number Program in BASIC
! (da 3 a SIZE*2) from Byte September 1981
!---------------------------------------------------
CONST SIZE%=8190
DIM FLAGS%[SIZE%]
BEGIN
PRINT("Only 1 iteration")
COUNT%=0
FOR I%=0 TO SIZE% DO
IF FLAGS%[I%]=TRUE THEN
!$NULL
ELSE
PRIME%=I%+I%+3
K%=I%+PRIME%
WHILE NOT (K%>SIZE%) DO
FLAGS%[K%]=TRUE
K%=K%+PRIME%
END WHILE
PRINT(PRIME%;)
COUNT%=COUNT%+1
END IF
END FOR
PRINT
PRINT(COUNT%;" PRIMES")
END PROGRAM
```
last lines of the output screen
```txt
15749 15761 15767 15773 15787 15791 15797 15803 15809 15817 15823
15859 15877 15881 15887 15889 15901 15907 15913 15919 15923 15937
15959 15971 15973 15991 16001 16007 16033 16057 16061 16063 16067
16069 16073 16087 16091 16097 16103 16111 16127 16139 16141 16183
16187 16189 16193 16217 16223 16229 16231 16249 16253 16267 16273
16301 16319 16333 16339 16349 16361 16363 16369 16381
1899 PRIMES
```
## Euphoria
```euphoria
constant limit = 1000
sequence flags,primes
flags = repeat(1, limit)
for i = 2 to sqrt(limit) do
if flags[i] then
for k = i*i to limit by i do
flags[k] = 0
end for
end if
end for
primes = {}
for i = 2 to limit do
if flags[i] = 1 then
primes &= i
end if
end for
? primes
```
Output:
```txt
{2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,
97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,
181,191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271,
277,281,283,293,307,311,313,317,331,337,347,349,353,359,367,373,379,
383,389,397,401,409,419,421,431,433,439,443,449,457,461,463,467,479,
487,491,499,503,509,521,523,541,547,557,563,569,571,577,587,593,599,
601,607,613,617,619,631,641,643,647,653,659,661,673,677,683,691,701,
709,719,727,733,739,743,751,757,761,769,773,787,797,809,811,821,823,
827,829,839,853,857,859,863,877,881,883,887,907,911,919,929,937,941,
947,953,967,971,977,983,991,997}
```
## F Sharp
### Short Sweet Functional and Idiotmatic
Well lists may not be lazy, but if you call it a sequence then it's a lazy list!
```fsharp
(*
An interesting implementation of The Sieve of Eratosthenes.
Nigel Galloway April 7th., 2017.
*)
let SofE =
let rec fn n g = seq{ match n with
|1 -> yield false; yield! fn g g
|_ -> yield true; yield! fn (n - 1) g}
let rec fg ng = seq {
let g = (Seq.findIndex(id) ng) + 2 // decreasingly inefficient with range at O(n)!
yield g; yield! fn (g - 1) g |> Seq.map2 (&&) ng |> Seq.cache |> fg }
Seq.initInfinite (fun x -> true) |> fg
```
```txt
> SofE |> Seq.take 10 |> Seq.iter(printfn "%d");;
2
3
5
7
11
13
17
19
23
29
```
Although interesting intellectually, and although the algorithm is more Sieve of Eratosthenes (SoE) than not in that it uses a progression of composite number representations separated by base prime gaps to cull, it isn't really SoE in performance due to several used functions that aren't linear with range, such as the "findIndex" that scans from the beginning of all primes to find the next un-culled value as the next prime in the sequence and the general slowness and inefficiency of F# nested sequence generation.
It is so slow that it takes in the order of seconds just to find the primes to a thousand!
For practical use, one would be much better served by any of the other functional sieves below, which can sieve to a million in less time than it takes this one to sieve to ten thousand. Those other functional sieves aren't all that many lines of code than this one.
### Functional
'''Richard Bird Sieve'''
This is the idea behind Richard Bird's unbounded code presented in the Epilogue of [http://www.cs.hmc.edu/~oneill/papers/Sieve-JFP.pdf M. O'Neill's article] in Haskell. It is about twice as much code as the Haskell code because F# does not have a built-in lazy list so that the effect must be constructed using a Co-Inductive Stream (CIS) type since no memoization is required, along with the use of recursive functions in combination with sequences. The type inference needs some help with the new CIS type (including selecting the generic type for speed). Note the use of recursive functions to implement multiple non-sharing delayed generating base primes streams, which along with these being non-memoizing means that the entire primes stream is not held in memory as for the original Bird code:
```fsharp
type 'a CIS = CIS of 'a * (unit -> 'a CIS) //'Co Inductive Stream for laziness
let primesBird() =
let rec (^^) (CIS(x, xtlf) as xs) (CIS(y, ytlf) as ys) = // stream merge function
if x < y then CIS(x, fun() -> xtlf() ^^ ys)
elif y < x then CIS(y, fun() -> xs ^^ ytlf())
else CIS(x, fun() -> xtlf() ^^ ytlf()) // no duplication
let pmltpls p = let rec nxt c = CIS(c, fun() -> nxt (c + p)) in nxt (p * p)
let rec allmltps (CIS(p, ptlf)) = CIS(pmltpls p, fun() -> allmltps (ptlf()))
let rec cmpsts (CIS(CIS(c, ctlf), amstlf)) =
CIS(c, fun() -> (ctlf()) ^^ (cmpsts (amstlf())))
let rec minusat n (CIS(c, ctlf) as cs) =
if n < c then CIS(n, fun() -> minusat (n + 1u) cs)
else minusat (n + 1u) (ctlf())
let rec baseprms() = CIS(2u, fun() -> baseprms() |> allmltps |> cmpsts |> minusat 3u)
Seq.unfold (fun (CIS(p, ptlf)) -> Some(p, ptlf())) (baseprms())
```
The above code sieves all numbers of two and up including all even numbers as per the page specification; the following code makes the very minor changes for an odds-only sieve, with a speedup of over a factor of two:
```fsharp
type 'a CIS = CIS of 'a * (unit -> 'a CIS) //'Co Inductive Stream for laziness
let primesBirdOdds() =
let rec (^^) (CIS(x, xtlf) as xs) (CIS(y, ytlf) as ys) = // stream merge function
if x < y then CIS(x, fun() -> xtlf() ^^ ys)
elif y < x then CIS(y, fun() -> xs ^^ ytlf())
else CIS(x, fun() -> xtlf() ^^ ytlf()) // no duplication
let pmltpls p = let adv = p + p
let rec nxt c = CIS(c, fun() -> nxt (c + adv)) in nxt (p * p)
let rec allmltps (CIS(p, ptlf)) = CIS(pmltpls p, fun() -> allmltps (ptlf()))
let rec cmpsts (CIS(CIS(c, ctlf), amstlf)) =
CIS(c, fun() -> ctlf() ^^ cmpsts (amstlf()))
let rec minusat n (CIS(c, ctlf) as cs) =
if n < c then CIS(n, fun() -> minusat (n + 2u) cs)
else minusat (n + 2u) (ctlf())
let rec oddprms() = CIS(3u, fun() -> oddprms() |> allmltps |> cmpsts |> minusat 5u)
Seq.unfold (fun (CIS(p, ptlf)) -> Some(p, ptlf())) (CIS(2u, fun() -> oddprms()))
```
'''Tree Folding Sieve'''
The above code is still somewhat inefficient as it operates on a linear right extending structure that deepens linearly with increasing base primes (those up to the square root of the currently sieved number); the following code changes the structure into an infinite binary tree-like folding by combining each pair of prime composite streams before further processing as usual - this decreases the processing by approximately a factor of log n:
```fsharp
type 'a CIS = CIS of 'a * (unit -> 'a CIS) //'Co Inductive Stream for laziness
let primesTreeFold() =
let rec (^^) (CIS(x, xtlf) as xs) (CIS(y, ytlf) as ys) = // stream merge function
if x < y then CIS(x, fun() -> xtlf() ^^ ys)
elif y < x then CIS(y, fun() -> xs ^^ ytlf())
else CIS(x, fun() -> xtlf() ^^ ytlf()) // no duplication
let pmltpls p = let adv = p + p
let rec nxt c = CIS(c, fun() -> nxt (c + adv)) in nxt (p * p)
let rec allmltps (CIS(p, ptlf)) = CIS(pmltpls p, fun() -> allmltps (ptlf()))
let rec pairs (CIS(cs0, cs0tlf)) =
let (CIS(cs1, cs1tlf)) = cs0tlf() in CIS(cs0 ^^ cs1, fun() -> pairs (cs1tlf()))
let rec cmpsts (CIS(CIS(c, ctlf), amstlf)) =
CIS(c, fun() -> ctlf() ^^ (cmpsts << pairs << amstlf)())
let rec minusat n (CIS(c, ctlf) as cs) =
if n < c then CIS(n, fun() -> minusat (n + 2u) cs)
else minusat (n + 2u) (ctlf())
let rec oddprms() = CIS(3u, fun() -> oddprms() |> allmltps |> cmpsts |> minusat 5u)
Seq.unfold (fun (CIS(p, ptlf)) -> Some(p, ptlf())) (CIS(2u, fun() -> oddprms()))
```
The above code is over four times faster than the "BirdOdds" version (at least 10x faster than the first, "primesBird", producing the millionth prime) and is moderately useful for a range of the first million primes or so.
'''Priority Queue Sieve'''
In order to investigate Priority Queue Sieves as espoused by O'Neill in the referenced article, one must find an equivalent implementation of a Min Heap Priority Queue as used by her. There is such an purely functional implementation [http://rosettacode.org/wiki/Priority_queue#Functional in RosettaCode translated from the Haskell code she used], from which the essential parts are duplicated here (Note that the key value is given an integer type in order to avoid the inefficiency of F# in generic comparison):
```fsharp>[ = struct val k:uint32 val v:'V new(k,v) = {k=k;v=v} end
[]
[]
type PQ<'V> =
| Mt
| Br of HeapEntry<'V> * PQ<'V> * PQ<'V>
let empty = Mt
let peekMin = function | Br(kv, _, _) -> Some(kv.k, kv.v)
| _ -> None
let rec push wk wv =
function | Mt -> Br(HeapEntry(wk, wv), Mt, Mt)
| Br(vkv, ll, rr) ->
if wk <= vkv.k then
Br(HeapEntry(wk, wv), push vkv.k vkv.v rr, ll)
else Br(vkv, push wk wv rr, ll)
let private siftdown wk wv pql pqr =
let rec sift pl pr =
match pl with
| Mt -> Br(HeapEntry(wk, wv), Mt, Mt)
| Br(vkvl, pll, plr) ->
match pr with
| Mt -> if wk <= vkvl.k then Br(HeapEntry(wk, wv), pl, Mt)
else Br(vkvl, Br(HeapEntry(wk, wv), Mt, Mt), Mt)
| Br(vkvr, prl, prr) ->
if wk <= vkvl.k && wk <= vkvr.k then Br(HeapEntry(wk, wv), pl, pr)
elif vkvl.k <= vkvr.k then Br(vkvl, sift pll plr, pr)
else Br(vkvr, pl, sift prl prr)
sift pql pqr
let replaceMin wk wv = function | Mt -> Mt
| Br(_, ll, rr) -> siftdown wk wv ll rr
```
Except as noted for any individual code, all of the following codes need the following prefix code in order to implement the non-memoizing Co-Inductive Streams (CIS's) and to set the type of particular constants used in the codes to the same time as the "Prime" type:
```fsharp
type CIS<'T> = struct val v: 'T val cont: unit -> CIS<'T> new(v,cont) = {v=v;cont=cont} end
type Prime = uint32
let frstprm = 2u
let frstoddprm = 3u
let inc1 = 1u
let inc = 2u
```
The F# equivalent to O'Neill's "odds-only" code is then implemented as follows, which needs the included changed prefix in order to change the primes type to a larger one to prevent overflow (as well the key type for the MinHeap needs to be changed from uint32 to uint64); it is functionally the same as the O'Neill code other than for minor changes to suit the use of CIS streams and the option output of the "peekMin" function:
```fsharp
type CIS<'T> = struct val v: 'T val cont: unit -> CIS<'T> new(v,cont) = {v=v;cont=cont} end
type Prime = uint64
let frstprm = 2UL
let frstoddprm = 3UL
let inc = 2UL
let primesPQ() =
let pmult p (xs: CIS) = // does map (* p) xs
let rec nxtm (cs: CIS) =
CIS(p * cs.v, fun() -> nxtm (cs.cont())) in nxtm xs
let insertprime p xs table =
MinHeap.push (p * p) (pmult p xs) table
let rec sieve' (ns: CIS) table =
let nextcomposite = match MinHeap.peekMin table with
| None -> ns.v // never happens
| Some (k, _) -> k
let rec adjust table =
let (n, advs) = match MinHeap.peekMin table with
| None -> (ns.v, ns.cont()) // never happens
| Some kv -> kv
if n <= ns.v then adjust (MinHeap.replaceMin advs.v (advs.cont()) table)
else table
if nextcomposite <= ns.v then sieve' (ns.cont()) (adjust table)
else let n = ns.v in CIS(n, fun() ->
let nxtns = ns.cont() in sieve' nxtns (insertprime n nxtns table))
let rec sieve (ns: CIS) = let n = ns.v in CIS(n, fun() ->
let nxtns = ns.cont() in sieve' nxtns (insertprime n nxtns MinHeap.empty))
let odds = // is the odds CIS from 3 up
let rec nxto i = CIS(i, fun() -> nxto (i + inc)) in nxto frstoddprm
Seq.unfold (fun (cis: CIS) -> Some(cis.v, cis.cont()))
(CIS(frstprm, fun() -> (sieve odds)))
```
However, that algorithm suffers in speed and memory use due to over-eager adding of prime composite streams to the queue such that the queue used is much larger than it needs to be and a much larger range of primes number must be used in order to avoid numeric overflow on the square of the prime added to the queue. The following code corrects that by using a secondary (actually a multiple of) base primes streams which are constrained to be based on a prime that is no larger than the square root of the currently sieved number - this permits the use of much smaller Prime types as per the default prefix:
```fsharp
let primesPQx() =
let rec nxtprm n pq q (bps: CIS) =
if n >= q then let bp = bps.v in let adv = bp + bp
let nbps = bps.cont() in let nbp = nbps.v
nxtprm (n + inc) (MinHeap.push (n + adv) adv pq) (nbp * nbp) nbps
else let ck, cv = match MinHeap.peekMin pq with
| None -> (q, inc) // only happens until first insertion
| Some kv -> kv
if n >= ck then let rec adjpq ck cv pq =
let npq = MinHeap.replaceMin (ck + cv) cv pq
match MinHeap.peekMin npq with
| None -> npq // never happens
| Some(nk, nv) -> if n >= nk then adjpq nk nv npq
else npq
nxtprm (n + inc) (adjpq ck cv pq) q bps
else CIS(n, fun() -> nxtprm (n + inc) pq q bps)
let rec oddprms() = CIS(frstoddprm, fun() ->
nxtprm (frstoddprm + inc) MinHeap.empty (frstoddprm * frstoddprm) (oddprms()))
Seq.unfold (fun (cis: CIS) -> Some(cis.v, cis.cont()))
(CIS(frstprm, fun() -> (oddprms())))
```
The above code is well over five times faster than the previous translated O'Neill version for the given variety of reasons.
Although slightly faster than the Tree Folding code, this latter code is also limited in practical usefulness to about the first one to ten million primes or so.
All of the above codes can be tested in the F# REPL with the following to produce the millionth prime (the "nth" function is zero based):
```txt
> primesXXX() |> Seq.nth 999999;;
```
where primesXXX() is replaced by the given primes generating function to be tested, and which all produce the following output (after a considerable wait in some cases):
```txt
val it : Prime = 15485863u
```
### Imperative
The following code is written in functional style other than it uses a mutable bit array to sieve the composites:
```fsharp
let primes limit =
let buf = System.Collections.BitArray(int limit + 1, true)
let cull p = { p * p .. p .. limit } |> Seq.iter (fun c -> buf.[int c] <- false)
{ 2u .. uint32 (sqrt (double limit)) } |> Seq.iter (fun c -> if buf.[int c] then cull c)
{ 2u .. limit } |> Seq.map (fun i -> if buf.[int i] then i else 0u) |> Seq.filter ((<>) 0u)
[]
let main argv =
if argv = null || argv.Length = 0 then failwith "no command line argument for limit!!!"
printfn "%A" (primes (System.UInt32.Parse argv.[0]) |> Seq.length)
0 // return an integer exit code
```
Substituting the following minor changes to the code for the "primes" function will only deal with the odd prime candidates for a speed up of over a factor of two as well as a reduction of the buffer size by a factor of two:
```fsharp
let primes limit =
let lmtb,lmtbsqrt = (limit - 3u) / 2u, (uint32 (sqrt (double limit)) - 3u) / 2u
let buf = System.Collections.BitArray(int lmtb + 1, true)
let cull i = let p = i + i + 3u in let s = p * (i + 1u) + i in
{ s .. p .. lmtb } |> Seq.iter (fun c -> buf.[int c] <- false)
{ 0u .. lmtbsqrt } |> Seq.iter (fun i -> if buf.[int i] then cull i )
let oddprms = { 0u .. lmtb } |> Seq.map (fun i -> if buf.[int i] then i + i + 3u else 0u)
|> Seq.filter ((<>) 0u)
seq { yield 2u; yield! oddprms }
```
The following code uses other functional forms for the inner culling loops of the "primes function" to reduce the use of inefficient sequences so as to reduce the execution time by another factor of almost three:
```fsharp
let primes limit =
let lmtb,lmtbsqrt = (limit - 3u) / 2u, (uint32 (sqrt (double limit)) - 3u) / 2u
let buf = System.Collections.BitArray(int lmtb + 1, true)
let rec culltest i = if i <= lmtbsqrt then
let p = i + i + 3u in let s = p * (i + 1u) + i in
let rec cullp c = if c <= lmtb then buf.[int c] <- false; cullp (c + p)
(if buf.[int i] then cullp s); culltest (i + 1u) in culltest 0u
seq {yield 2u; for i = 0u to lmtb do if buf.[int i] then yield i + i + 3u }
```
Now much of the remaining execution time is just the time to enumerate the primes as can be seen by turning "primes" into a primes counting function by substituting the following for the last line in the above code doing the enumeration; this makes the code run about a further five times faster:
```fsharp
let rec count i acc =
if i > int lmtb then acc else if buf.[i] then count (i + 1) (acc + 1) else count (i + 1) acc
count 0 1
```
Since the final enumeration of primes is the main remaining bottleneck, it is worth using a "roll-your-own" enumeration implemented as an object expression so as to save many inefficiencies in the use of the built-in seq computational expression by substituting the following code for the last line of the previous codes, which will decrease the execution time by a factor of over three (instead of almost five for the counting-only version, making it almost as fast):
```fsharp
let nmrtr() =
let i = ref -2
let rec nxti() = i:=!i + 1;if !i <= int lmtb && not buf.[!i] then nxti() else !i <= int lmtb
let inline curr() = if !i < 0 then (if !i= -1 then 2u else failwith "Enumeration not started!!!")
else let v = uint32 !i in v + v + 3u
{ new System.Collections.Generic.IEnumerator<_> with
member this.Current = curr()
interface System.Collections.IEnumerator with
member this.Current = box (curr())
member this.MoveNext() = if !i< -1 then i:=!i+1;true else nxti()
member this.Reset() = failwith "IEnumerator.Reset() not implemented!!!"a
interface System.IDisposable with
member this.Dispose() = () }
{ new System.Collections.Generic.IEnumerable<_> with
member this.GetEnumerator() = nmrtr()
interface System.Collections.IEnumerable with
member this.GetEnumerator() = nmrtr() :> System.Collections.IEnumerator }
```
The various optimization techniques shown here can be used "jointly and severally" on any of the basic versions for various trade-offs between code complexity and performance. Not shown here are other techniques of making the sieve faster, including extending wheel factorization to much larger wheels such as 2/3/5/7, pre-culling the arrays, page segmentation, and multi-processing.
### Almost functional Unbounded
the following '''odds-only''' implmentations are written in an almost functional style avoiding the use of mutability except for the contents of the data structures uses to hold the state of the and any mutability necessary to implement a "roll-your-own" IEnumberable iterator interface for speed.
'''Unbounded Dictionary (Mutable Hash Table) Based Sieve'''
The following code uses the DotNet Dictionary class instead of the above functional Priority Queue to implement the sieve; as average (amortized) hash table access is O(1) rather than O(log n) as for the priority queue, this implementation is slightly faster than the priority queue version for the first million primes and will always be faster for any range above some low range value:
```fsharp
type Prime = uint32
let frstprm = 2u
let frstoddprm = 3u
let inc = 2u
let primesDict() =
let dct = System.Collections.Generic.Dictionary()
let rec nxtprm n q (bps: CIS) =
if n >= q then let bp = bps.v in let adv = bp + bp
let nbps = bps.cont() in let nbp = nbps.v
dct.Add(n + adv, adv)
nxtprm (n + inc) (nbp * nbp) nbps
else if dct.ContainsKey(n) then
let adv = dct.[n]
dct.Remove(n) |> ignore
// let mutable nn = n + adv // ugly imperative code
// while dct.ContainsKey(nn) do nn <- nn + adv
// dct.Add(nn, adv)
let rec nxtmt k = // advance to next empty spot
if dct.ContainsKey(k) then nxtmt (k + adv)
else dct.Add(k, adv) in nxtmt (n + adv)
nxtprm (n + inc) q bps
else CIS(n, fun() -> nxtprm (n + inc) q bps)
let rec oddprms() = CIS(frstoddprm, fun() ->
nxtprm (frstoddprm + inc) (frstoddprm * frstoddprm) (oddprms()))
Seq.unfold (fun (cis: CIS) -> Some(cis.v, cis.cont()))
(CIS(frstprm, fun() -> (oddprms())))
```
The above code uses functional forms of code (with the imperative style commented out to show how it could be done imperatively) and also uses a recursive non-sharing secondary source of base primes just as for the Priority Queue version. As for the functional codes, the Primes type can easily be changed to "uint64" for wider range of sieving.
In spite of having true O(n log log n) Sieve of Eratosthenes computational complexity where n is the range of numbers to be sieved, the above code is still not particularly fast due to the time required to compute the hash values and manipulations of the hash table.
'''Unbounded Page-Segmented Bit-Packed Odds-Only Mutable Array Sieve'''
Note that the following code is used for the F# entry [[Extensible_prime_generator#Unbounded_Mutable_Array_Generator]] of the Extensible prime generator page.
All of the above unbounded implementations including the above Dictionary based version are quite slow due to their large constant factor computational overheads, making them more of an intellectual exercise than something practical, especially when larger sieving ranges are required. The following code implements an unbounded page segmented version of the sieve in not that many more lines of code, yet runs about 25 times faster than the Dictionary version for larger ranges of sieving such as to one billion; it uses functional forms without mutability other than for the contents of the arrays and the `primes` enumeration generator function that must use mutability for speed:
```fsharp
type Prime = float // use uint64/int64 for regular 64-bit F#
type private PrimeNdx = float // they are slow in JavaScript polyfills
let inline private prime n = float n // match these convenience conversions
let inline private primendx n = float n // with the types above!
let private cPGSZBTS = (1 <<< 14) * 8 // sieve buffer size in bits = CPUL1CACHE
type private SieveBuffer = uint8[]
/// a Co-Inductive Stream (CIS) of an "infinite" non-memoized series...
type private CIS<'T> = CIS of 'T * (unit -> CIS<'T>) //' apostrophe formatting adjustment
/// lazy list (memoized) series of base prime page arrays...
type private BasePrime = uint32
type private BasePrimeArr = BasePrime[]
type private BasePrimeArrs = BasePrimeArrs of BasePrimeArr * Option>
/// Masking array is faster than bit twiddle bit shifts!
let private cBITMASK = [| 1uy; 2uy; 4uy; 8uy; 16uy; 32uy; 64uy; 128uy |]
let private cullSieveBuffer lwi (bpas: BasePrimeArrs) (sb: SieveBuffer) =
let btlmt = (sb.Length <<< 3) - 1 in let lmti = lwi + primendx btlmt
let rec loopbp (BasePrimeArrs(bpa, bpatl) as ibpas) i =
if i >= bpa.Length then
match bpatl with
| None -> ()
| Some lv -> loopbp lv.Value 0 else
let bp = prime bpa.[i] in let bpndx = primendx ((bp - prime 3) / prime 2)
let s = (bpndx * primendx 2) * (bpndx + primendx 3) + primendx 3 in let bpint = int bp
if s <= lmti then
let s0 = // page cull start address calculation...
if s >= lwi then int (s - lwi) else
let r = (lwi - s) % (primendx bp)
if r = primendx 0 then 0 else int (bp - prime r)
let slmt = min btlmt (s0 - 1 + (bpint <<< 3))
let rec loopc c = // loop "unpeeling" is used so
if c <= slmt then // a constant mask can be used over the inner loop
let msk = cBITMASK.[c &&& 7]
let rec loopw w =
if w < sb.Length then sb.[w] <- sb.[w] ||| msk; loopw (w + bpint)
loopw (c >>> 3); loopc (c + bpint)
loopc s0; loopbp ibpas (i + 1) in loopbp bpas 0
/// fast Counting Look Up Table (CLUT) for pop counting...
let private cCLUT =
let arr = Array.zeroCreate 65536
let rec popcnt n cnt = if n > 0 then popcnt (n &&& (n - 1)) (cnt + 1) else uint8 cnt
let rec loop i = if i < 65536 then arr.[i] <- popcnt i 0; loop (i + 1)
loop 0; arr
let countSieveBuffer ndxlmt (sb: SieveBuffer): int =
let lstw = (ndxlmt >>> 3) &&& -2
let msk = (-2 <<< (ndxlmt &&& 15)) &&& 0xFFFF
let inline cntem i m =
int cCLUT.[int (((uint32 sb.[i + 1]) <<< 8) + uint32 sb.[i]) ||| m]
let rec loop i cnt =
if i >= lstw then cnt - cntem lstw msk else loop (i + 2) (cnt - cntem i 0)
loop 0 ((lstw <<< 3) + 16)
/// a CIS series of pages from the given start index with the given SieveBuffer size,
/// and provided with a polymorphic converter function to produce
/// and type of result from the culled page parameters...
let rec private makePrimePages strtwi btsz
(cnvrtrf: PrimeNdx -> SieveBuffer -> 'T): CIS<'T> =
let bpas = makeBasePrimes() in let sb = Array.zeroCreate (btsz >>> 3)
let rec nxtpg lwi =
Array.fill sb 0 sb.Length 0uy; cullSieveBuffer lwi bpas sb
CIS(cnvrtrf lwi sb, fun() -> nxtpg (lwi + primendx btsz))
nxtpg strtwi
/// secondary feed of lazy list of memoized pages of base primes...
and private makeBasePrimes(): BasePrimeArrs =
let sb2bpa lwi (sb: SieveBuffer) =
let bsbp = uint32 (primendx 3 + lwi + lwi)
let arr = Array.zeroCreate <| countSieveBuffer 255 sb
let rec loop i j =
if i < 256 then
if sb.[i >>> 3] &&& cBITMASK.[i &&& 7] <> 0uy then loop (i + 1) j
else arr.[j] <- bsbp + uint32 (i + i); loop (i + 1) (j + 1)
loop 0 0; arr
// finding the first page as not part of the loop and making succeeding
// pages lazy breaks the recursive data race!
let frstsb = Array.zeroCreate 32
let fkbpas = BasePrimeArrs(sb2bpa (primendx 0) frstsb, None)
cullSieveBuffer (primendx 0) fkbpas frstsb
let rec nxtbpas (CIS(bpa, tlf)) = BasePrimeArrs(bpa, Some(lazy (nxtbpas (tlf()))))
BasePrimeArrs(sb2bpa (primendx 0) frstsb,
Some(lazy (nxtbpas <| makePrimePages (primendx 256) 256 sb2bpa)))
/// produces a generator of primes; uses mutability for better speed...
let primes(): unit -> Prime =
let sb2prms lwi (sb: SieveBuffer) = lwi, sb in let mutable ndx = -1
let (CIS((nlwi, nsb), npgtlf)) = // use page generator function above!
makePrimePages (primendx 0) cPGSZBTS sb2prms
let mutable lwi = nlwi in let mutable sb = nsb
let mutable pgtlf = npgtlf
let mutable baseprm = prime 3 + prime (lwi + lwi)
fun() ->
if ndx < 0 then ndx <- 0; prime 2 else
let inline notprm i = sb.[i >>> 3] &&& cBITMASK.[i &&& 7] <> 0uy
while ndx < cPGSZBTS && notprm ndx do ndx <- ndx + 1
if ndx >= cPGSZBTS then // get next page if over
let (CIS((nlwi, nsb), npgtlf)) = pgtlf() in ndx <- 0
lwi <- nlwi; sb <- nsb; pgtlf <- npgtlf
baseprm <- prime 3 + prime (lwi + lwi)
while notprm ndx do ndx <- ndx + 1
let ni = ndx in ndx <- ndx + 1 // ready for next call!
baseprm + prime (ni + ni)
let countPrimesTo (limit: Prime): int = // much faster!
if limit < prime 3 then (if limit < prime 2 then 0 else 1) else
let topndx = (limit - prime 3) / prime 2 |> primendx
let sb2cnt lwi (sb: SieveBuffer) =
let btlmt = (sb.Length <<< 3) - 1 in let lmti = lwi + primendx btlmt
countSieveBuffer
(if lmti < topndx then btlmt else int (topndx - lwi)) sb, lmti
let rec loop (CIS((cnt, nxti), tlf)) count =
if nxti < topndx then loop (tlf()) (count + cnt)
else count + cnt
loop <| makePrimePages (primendx 0) cPGSZBTS sb2cnt <| 1
/// sequences are convenient but slow...
let primesSeq() = primes() |> Seq.unfold (fun gen -> Some(gen(), gen))
printfn "The first 25 primes are: %s"
( primesSeq() |> Seq.take 25
|> Seq.fold (fun s p -> s + string p + " ") "" )
printfn "There are %d primes up to a million."
( primesSeq() |> Seq.takeWhile ((>=) (prime 1000000)) |> Seq.length )
let rec cntto gen lmt cnt = // faster than seq's but still slow
if gen() > lmt then cnt else cntto gen lmt (cnt + 1)
let limit = prime 1_000_000_000
let start = System.DateTime.Now.Ticks
// let answr = cntto (primes()) limit 0 // slower way!
let answr = countPrimesTo limit // over twice as fast way!
let elpsd = (System.DateTime.Now.Ticks - start) / 10000L
printfn "Found %d primes to %A in %d milliseconds." answr limit elpsd
```
```txt
The first 25 primes are: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
There are 78498 primes up to a million.
Found 50847534 primes to 1000000000 in 2161 milliseconds.
```
As with all of the efficient unbounded sieves, the above code uses a secondary enumerator of the base primes less than the square root of the currently culled range, which is this case is a lazy (deferred memoized evaluation) binding by small pages of base primes which also uses the laziness of the deferral of subsequent pages so as to avoid a race condition.
The above code is written to output the "uint64" type for very large ranges of primes since there is little computational cost to doing this for this algorithm when used with 64-bit compilation; however, for the Fable transpiled to JavaScript, the largest contiguous integer that can be represented is the 64-bit floating point mantissa of 52 bits and thus the large numbers can be represented by floats in this case since a 64-bit polyfill is very slow. As written, the practical range for this sieve is about 16 billion, however, it can be extended to about 10^14 (a week or two of execution time) by setting the "PGSZBTS" constant to the size of the CPU L2 cache rather than the L1 cache (L2 is up to about two Megabytes for modern high end desktop CPU's) at a slight loss of efficiency (a factor of up to two or so) per composite number culling operation due to the slower memory access time. When the Fable compilation option is used, execution speed is roughly the same as using F# with DotNet Core.
Even with the custom `primes` enumerator generator (the F#/Fable built-in sequence operators are terribly inefficient), the time to enumerate the resulting primes takes longer than the time to actually cull the composite numbers from the sieving arrays. The time to do the actual culling is thus over 50 times faster than done using the Dictionary version. The slowness of enumeration, no matter what further tweaks are done to improve it (each value enumerated will always take a function calls and a scan loop that will always take something in the order of 100 CPU clock cycles per value), means that further gains in speed using extreme wheel factorization and multi-processing have little point unless the actual work on the resulting primes is done through use of auxiliary functions not using iteration. Such a function is provided here to count the primes by pages using a "pop count" look up table to reduce the counting time to only a small fraction of a second.
## Factor
Factor already contains two implementations of the sieve of Eratosthenes in math.primes.erato and math.primes.erato.fast. It is suggested to use one of them for real use, as they use faster types, faster unsafe arithmetic, and/or wheels to speed up the sieve further. Shown here is a more straightforward implementation that adheres to the restrictions given by the task (namely, no wheels).
Factor is pleasantly multiparadigm. Usually, it's natural to write more functional or declarative code in Factor, but this is an instance where it is more natural to write imperative code. Lexical variables are useful here for expressing the necessary mutations in a clean way.
```factor
USING: bit-arrays io kernel locals math math.functions
math.ranges prettyprint sequences ;
IN: rosetta-code.sieve-of-erato
dup set-bits ?{ f f } prepend ;
! Given the sieve and a prime starting index, create a range of
! values to mark composite. Start at the square of the prime.
: to-mark ( seq n -- range )
[ length 1 - ] [ dup dup * ] bi* -rot ;
! Mark multiples of prime n as composite.
: mark-nths ( seq n -- )
dupd to-mark [ swap [ f ] 2dip set-nth ] with each ;
: next-prime ( index seq -- n ) [ t = ] find-from drop ;
PRIVATE>
:: sieve ( n -- seq )
n sqrt 2 n init-sieve :> ( limit i! s )
[ i limit < ] ! sqrt optimization
[ s i mark-nths i 1 + s next-prime i! ] while t s indices ;
: sieve-demo ( -- )
"Primes up to 120 using sieve of Eratosthenes:" print
120 sieve . ;
MAIN: sieve-demo
```
=={{header|Fōrmulæ}}==
In [http://wiki.formulae.org/Sieve_of_Eratosthenes this] page you can see the solution of this task.
Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text ([http://wiki.formulae.org/Editing_F%C5%8Drmul%C3%A6_expressions more info]). Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for transportation effects more than visualization and edition.
The option to show Fōrmulæ programs and their results is showing images. Unfortunately images cannot be uploaded in Rosetta Code.
## Forth
: prime? ( n -- ? ) here + c@ 0= ;
: composite! ( n -- ) here + 1 swap c! ;
: sieve ( n -- )
here over erase
2
begin
2dup dup * >
while
dup prime? if
2dup dup * do
i composite!
dup +loop
then
1+
repeat
drop
." Primes: " 2 do i prime? if i . then loop ;
100 sieve
## Fortran
```fortran
program sieve
implicit none
integer, parameter :: i_max = 100
integer :: i
logical, dimension (i_max) :: is_prime
is_prime = .true.
is_prime (1) = .false.
do i = 2, int (sqrt (real (i_max)))
if (is_prime (i)) is_prime (i * i : i_max : i) = .false.
end do
do i = 1, i_max
if (is_prime (i)) write (*, '(i0, 1x)', advance = 'no') i
end do
write (*, *)
end program sieve
```
Output:
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
```
Optimised using a pre-computed wheel based on 2:
```fortran
program sieve_wheel_2
implicit none
integer, parameter :: i_max = 100
integer :: i
logical, dimension (i_max) :: is_prime
is_prime = .true.
is_prime (1) = .false.
is_prime (4 : i_max : 2) = .false.
do i = 3, int (sqrt (real (i_max))), 2
if (is_prime (i)) is_prime (i * i : i_max : 2 * i) = .false.
end do
do i = 1, i_max
if (is_prime (i)) write (*, '(i0, 1x)', advance = 'no') i
end do
write (*, *)
end program sieve_wheel_2
```
Output:
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
```
## FreeBASIC
```freebasic
' FB 1.05.0
Sub sieve(n As Integer)
If n < 2 Then Return
Dim a(2 To n) As Integer
For i As Integer = 2 To n : a(i) = i : Next
Dim As Integer p = 2, q
' mark non-prime numbers by setting the corresponding array element to 0
Do
For j As Integer = p * p To n Step p
a(j) = 0
Next j
' look for next non-zero element in array after 'p'
q = 0
For j As Integer = p + 1 To Sqr(n)
If a(j) <> 0 Then
q = j
Exit For
End If
Next j
If q = 0 Then Exit Do
p = q
Loop
' print the non-zero numbers remaining i.e. the primes
For i As Integer = 2 To n
If a(i) <> 0 Then
Print Using "####"; a(i);
End If
Next
Print
End Sub
Print "The primes up to 1000 are :"
Print
sieve(1000)
Print
Print "Press any key to quit"
Sleep
```
```txt
The primes up to 1000 are :
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71
73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173
179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281
283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409
419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541
547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659
661 673 677 683 691 701 709 719 727 733 739 743 751 757 761 769 773 787 797 809
811 821 823 827 829 839 853 857 859 863 877 881 883 887 907 911 919 929 937 941
947 953 967 971 977 983 991 997
```
## Free Pascal
### Basic version
function Sieve returns a list of primes less than or equal to the given aLimit
```pascal
program prime_sieve;
{$mode objfpc}{$coperators on}
uses
SysUtils, GVector;
type
TPrimeList = specialize TVector;
function Sieve(aLimit: DWord): TPrimeList;
var
IsPrime: array of Boolean;
I, SqrtBound: DWord;
J: QWord;
begin
Result := TPrimeList.Create;
Inc(aLimit, Ord(aLimit < High(DWord))); //not a problem because High(DWord) is composite
SetLength(IsPrime, aLimit);
FillChar(Pointer(IsPrime)^, aLimit, Byte(True));
SqrtBound := Trunc(Sqrt(aLimit));
for I := 2 to aLimit do
if IsPrime[I] then
begin
Result.PushBack(I);
if I <= SqrtBound then
begin
J := I * I;
repeat
IsPrime[J] := False;
J += I;
until J > aLimit;
end;
end;
end;
//usage
var
Limit: DWord = 0;
function ReadLimit: Boolean;
var
Lim: Int64;
begin
if (ParamCount = 1) and Lim.TryParse(ParamStr(1), Lim) then
if (Lim >= 0) and (Lim <= High(DWord)) then
begin
Limit := DWord(Lim);
exit(True);
end;
Result := False;
end;
procedure PrintUsage;
begin
WriteLn('Usage: prime_sieve0 Limit');
WriteLn(' where Limit in the range [0, ', High(DWord), ']');
Halt;
end;
procedure PrintPrimes(aList: TPrimeList);
var
I: DWord;
begin
for I := 0 to aList.Size - 2 do
Write(aList[I], ', ');
WriteLn(aList[aList.Size - 1]);
aList.Free;
end;
begin
if not ReadLimit then
PrintUsage;
try
PrintPrimes(Sieve(Limit));
except
on e: Exception do
WriteLn('An exception ', e.ClassName, ' occurred with message: ', e.Message);
end;
end.
```
===Alternative segmented(odds only) version===
function OddSegmentSieve returns a list of primes less than or equal to the given aLimit
```pascal
program prime_sieve;
{$mode objfpc}{$coperators on}
uses
SysUtils, Math;
type
TPrimeList = array of DWord;
function OddSegmentSieve(aLimit: DWord): TPrimeList;
function EstimatePrimeCount(aLimit: DWord): DWord;
begin
case aLimit of
0..1: Result := 0;
2..200: Result := Trunc(1.6 * aLimit/Ln(aLimit)) + 1;
else
Result := Trunc(aLimit/(Ln(aLimit) - 2)) + 1;
end;
end;
function Sieve(aLimit: DWord; aNeed2: Boolean): TPrimeList;
var
IsPrime: array of Boolean;
I: DWord = 3;
J, SqrtBound: DWord;
Count: Integer = 0;
begin
if aLimit < 2 then
exit(nil);
SetLength(IsPrime, (aLimit - 1) div 2);
FillChar(Pointer(IsPrime)^, Length(IsPrime), Byte(True));
SetLength(Result, EstimatePrimeCount(aLimit));
SqrtBound := Trunc(Sqrt(aLimit));
if aNeed2 then
begin
Result[0] := 2;
Inc(Count);
end;
for I := 0 to High(IsPrime) do
if IsPrime[I] then
begin
Result[Count] := I * 2 + 3;
if Result[Count] <= SqrtBound then
begin
J := Result[Count] * Result[Count];
repeat
IsPrime[(J - 3) div 2] := False;
J += Result[Count] * 2;
until J > aLimit;
end;
Inc(Count);
end;
SetLength(Result, Count);
end;
const
PAGE_SIZE = $8000;
var
IsPrime: array[0..Pred(PAGE_SIZE)] of Boolean; //current page
SmallPrimes: TPrimeList = nil;
I: QWord;
J, PageHigh, Prime: DWord;
Count: Integer;
begin
if aLimit < PAGE_SIZE div 4 then
exit(Sieve(aLimit, True));
I := Trunc(Sqrt(aLimit));
SmallPrimes := Sieve(I + 1, False);
Count := Length(SmallPrimes) + 1;
I += Ord(not Odd(I));
SetLength(Result, EstimatePrimeCount(aLimit));
while I <= aLimit do
begin
PageHigh := Min(Pred(PAGE_SIZE * 2), aLimit - I);
FillChar(IsPrime, PageHigh div 2 + 1, Byte(True));
for Prime in SmallPrimes do
begin
J := DWord(I) mod Prime;
if J <> 0 then
J := Prime shl (1 - J and 1) - J;
while J <= PageHigh do
begin
IsPrime[J div 2] := False;
J += Prime * 2;
end;
end;
for J := 0 to PageHigh div 2 do
if IsPrime[J] then
begin
Result[Count] := J * 2 + I;
Inc(Count);
end;
I += PAGE_SIZE * 2;
end;
SetLength(Result, Count);
Result[0] := 2;
Move(SmallPrimes[0], Result[1], Length(SmallPrimes) * SizeOf(DWord));
end;
//usage
var
Limit: DWord = 0;
function ReadLimit: Boolean;
var
Lim: Int64;
begin
if (ParamCount = 1) and Lim.TryParse(ParamStr(1), Lim) then
if (Lim >= 0) and (Lim <= High(DWord)) then
begin
Limit := DWord(Lim);
exit(True);
end;
Result := False;
end;
procedure PrintUsage;
begin
WriteLn('Usage: prime_sieve3 Limit');
WriteLn(' where Limit in the range [2, ', High(DWord), ']');
Halt;
end;
procedure PrintPrimes(const aList: TPrimeList);
var
I: DWord;
begin
for I := 0 to Length(aList) - 2 do
Write(aList[I], ', ');
WriteLn(aList[High(aList)]);
end;
begin
if not ReadLimit then
PrintUsage;
PrintPrimes(OddSegmentSieve(Limit));
end.
```
## Frink
```frink
n = eval[input["Enter highest number: "]]
results = array[sieve[n]]
println[results]
println[length[results] + " prime numbers less than or equal to " + n]
sieve[n] :=
{
// Initialize array
array = array[0 to n]
array@1 = 0
for i = 2 to ceil[sqrt[n]]
if array@i != 0
for j = i^2 to n step i
array@j = 0
return select[array, { |x| x != 0 }]
}
```
## FutureBasic
### Basic sieve of array of booleans
```futurebasic
include "ConsoleWindow"
begin globals
dim dynamic gPrimes(1) as Boolean
end globals
local fn SieveOfEratosthenes( n as long )
dim as long i, j
for i = 2 to n
for j = i * i to n step i
gPrimes(j) = _true
next
if gPrimes(i) = 0 then print i;
next i
kill gPrimes
end fn
fn SieveOfEratosthenes( 100 )
```
Output:
```txt
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
```
## GAP
```gap
Eratosthenes := function(n)
local a, i, j;
a := ListWithIdenticalEntries(n, true);
if n < 2 then
return [];
else
for i in [2 .. n] do
if a[i] then
j := i*i;
if j > n then
return Filtered([2 .. n], i -> a[i]);
else
while j <= n do
a[j] := false;
j := j + i;
od;
fi;
fi;
od;
fi;
end;
Eratosthenes(100);
[ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 ]
```
## GLBasic
```GLBasic
// Sieve of Eratosthenes (find primes)
// GLBasic implementation
GLOBAL n%, k%, limit%, flags%[]
limit = 100 // search primes up to this number
DIM flags[limit+1] // GLBasic arrays start at 0
FOR n = 2 TO SQR(limit)
IF flags[n] = 0
FOR k = n*n TO limit STEP n
flags[k] = 1
NEXT
ENDIF
NEXT
// Display the primes
FOR n = 2 TO limit
IF flags[n] = 0 THEN STDOUT n + ", "
NEXT
KEYWAIT
```
## Go
### Basic sieve of array of booleans
```go
package main
import "fmt"
func main() {
const limit = 201 // means sieve numbers < 201
// sieve
c := make([]bool, limit) // c for composite. false means prime candidate
c[1] = true // 1 not considered prime
p := 2
for {
// first allowed optimization: outer loop only goes to sqrt(limit)
p2 := p * p
if p2 >= limit {
break
}
// second allowed optimization: inner loop starts at sqr(p)
for i := p2; i < limit; i += p {
c[i] = true // it's a composite
}
// scan to get next prime for outer loop
for {
p++
if !c[p] {
break
}
}
}
// sieve complete. now print a representation.
for n := 1; n < limit; n++ {
if c[n] {
fmt.Print(" .")
} else {
fmt.Printf("%3d", n)
}
if n%20 == 0 {
fmt.Println("")
}
}
}
```
Output:
```txt
. 2 3 . 5 . 7 . . . 11 . 13 . . . 17 . 19 .
. . 23 . . . . . 29 . 31 . . . . . 37 . . .
41 . 43 . . . 47 . . . . . 53 . . . . . 59 .
61 . . . . . 67 . . . 71 . 73 . . . . . 79 .
. . 83 . . . . . 89 . . . . . . . 97 . . .
101 .103 . . .107 .109 . . .113 . . . . . . .
. . . . . .127 . . .131 . . . . .137 .139 .
. . . . . . . .149 .151 . . . . .157 . . .
. .163 . . .167 . . . . .173 . . . . .179 .
181 . . . . . . . . .191 .193 . . .197 .199 .
```
===Odds-only bit-packed array output-enumerating version===
The above version's output is rather specialized; the following version uses a closure function to enumerate over the culled composite number array, which is bit packed. By using this scheme for output, no extra memory is required above that required for the culling array:
```go
package main
import (
"fmt"
"math"
)
func primesOdds(top uint) func() uint {
topndx := int((top - 3) / 2)
topsqrtndx := (int(math.Sqrt(float64(top))) - 3) / 2
cmpsts := make([]uint, (topndx/32)+1)
for i := 0; i <= topsqrtndx; i++ {
if cmpsts[i>>5]&(uint(1)<<(uint(i)&0x1F)) == 0 {
p := (i << 1) + 3
for j := (p*p - 3) >> 1; j <= topndx; j += p {
cmpsts[j>>5] |= 1 << (uint(j) & 0x1F)
}
}
}
i := -1
return func() uint {
oi := i
if i <= topndx {
i++
}
for i <= topndx && cmpsts[i>>5]&(1<<(uint(i)&0x1F)) != 0 {
i++
}
if oi < 0 {
return 2
} else {
return (uint(oi) << 1) + 3
}
}
}
func main() {
iter := primesOdds(100)
for v := iter(); v <= 100; v = iter() {
print(v, " ")
}
iter = primesOdds(1000000)
count := 0
for v := iter(); v <= 1000000; v = iter() {
count++
}
fmt.Printf("\r\n%v\r\n", count)
}
```
```txt
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
78498
```
### Sieve Tree
A fairly odd sieve tree method:
```go
package main
import "fmt"
type xint uint64
type xgen func()(xint)
func primes() func()(xint) {
pp, psq := make([]xint, 0), xint(25)
var sieve func(xint, xint)xgen
sieve = func(p, n xint) xgen {
m, next := xint(0), xgen(nil)
return func()(r xint) {
if next == nil {
r = n
if r <= psq {
n += p
return
}
next = sieve(pp[0] * 2, psq) // chain in
pp = pp[1:]
psq = pp[0] * pp[0]
m = next()
}
switch {
case n < m: r, n = n, n + p
case n > m: r, m = m, next()
default: r, n, m = n, n + p, next()
}
return
}
}
f := sieve(6, 9)
n, p := f(), xint(0)
return func()(xint) {
switch {
case p < 2: p = 2
case p < 3: p = 3
default:
for p += 2; p == n; {
p += 2
if p > n {
n = f()
}
}
pp = append(pp, p)
}
return p
}
}
func main() {
for i, p := 0, primes(); i < 100000; i++ {
fmt.Println(p())
}
}
```
===Concurrent Daisy-chain sieve===
A concurrent prime sieve adopted from the example in the "Go Playground" window at http://golang.org/
```go
package main
import "fmt"
// Send the sequence 2, 3, 4, ... to channel 'out'
func Generate(out chan<- int) {
for i := 2; ; i++ {
out <- i // Send 'i' to channel 'out'
}
}
// Copy the values from 'in' channel to 'out' channel,
// removing the multiples of 'prime' by counting.
// 'in' is assumed to send increasing numbers
func Filter(in <-chan int, out chan<- int, prime int) {
m := prime + prime // first multiple of prime
for {
i := <- in // Receive value from 'in'
for i > m {
m = m + prime // next multiple of prime
}
if i < m {
out <- i // Send 'i' to 'out'
}
}
}
// The prime sieve: Daisy-chain Filter processes
func Sieve(out chan<- int) {
gen := make(chan int) // Create a new channel
go Generate(gen) // Launch Generate goroutine
for {
prime := <- gen
out <- prime
ft := make(chan int)
go Filter(gen, ft, prime)
gen = ft
}
}
func main() {
sv := make(chan int) // Create a new channel
go Sieve(sv) // Launch Sieve goroutine
for i := 0; i < 1000; i++ {
prime := <- sv
if i >= 990 {
fmt.Printf("%4d ", prime)
if (i+1)%20==0 {
fmt.Println("")
}
}
}
}
```
The output:
```txt
7841 7853 7867 7873 7877 7879 7883 7901 7907 7919
```
[http://ideone.com/ixhHNO Runs at ~ n^2.1] empirically, producing up to n=3000 primes in under 5 seconds.
===Postponed Concurrent Daisy-chain sieve===
Here we postpone the ''creation'' of filters until the prime's square is seen in the input, to radically reduce the amount of filter channels in the sieve chain.
```go
package main
import "fmt"
// Send the sequence 2, 3, 4, ... to channel 'out'
func Generate(out chan<- int) {
for i := 2; ; i++ {
out <- i // Send 'i' to channel 'out'
}
}
// Copy the values from 'in' channel to 'out' channel,
// removing the multiples of 'prime' by counting.
// 'in' is assumed to send increasing numbers
func Filter(in <-chan int, out chan<- int, prime int) {
m := prime * prime // start from square of prime
for {
i := <- in // Receive value from 'in'
for i > m {
m = m + prime // next multiple of prime
}
if i < m {
out <- i // Send 'i' to 'out'
}
}
}
// The prime sieve: Postponed-creation Daisy-chain of Filters
func Sieve(out chan<- int) {
gen := make(chan int) // Create a new channel
go Generate(gen) // Launch Generate goroutine
p := <- gen
out <- p
p = <- gen // make recursion shallower ---->
out <- p // (Go channels are _push_, not _pull_)
base_primes := make(chan int) // separate primes supply
go Sieve(base_primes)
bp := <- base_primes // 2 <---- here
bq := bp * bp // 4
for {
p = <- gen
if p == bq { // square of a base prime
ft := make(chan int)
go Filter(gen, ft, bp) // filter multiples of bp in gen out
gen = ft
bp = <- base_primes // 3
bq = bp * bp // 9
} else {
out <- p
}
}
}
func main() {
sv := make(chan int) // Create a new channel
go Sieve(sv) // Launch Sieve goroutine
lim := 25000
for i := 0; i < lim; i++ {
prime := <- sv
if i >= (lim-10) {
fmt.Printf("%4d ", prime)
if (i+1)%20==0 {
fmt.Println("")
}
}
}
}
```
The output:
```txt
286999 287003 287047 287057 287059 287087 287093 287099 287107 287117
```
[http://ideone.com/I0AXf5 Runs at ~ n^1.2] empirically, producing up to n=25,000 primes on ideone in under 5 seconds.
===Incremental Odds-only Sieve===
Uses Go's built-in hash tables to store odd composites, and defers adding new known composites until the square is seen.
```go
package main
import "fmt"
func main() {
primes := make(chan int)
go PrimeSieve(primes)
p := <-primes
for p < 100 {
fmt.Printf("%d ", p)
p = <-primes
}
fmt.Println()
}
func PrimeSieve(out chan int) {
out <- 2
out <- 3
primes := make(chan int)
go PrimeSieve(primes)
var p int
p = <-primes
p = <-primes
sieve := make(map[int]int)
q := p * p
n := p
for {
n += 2
step, isComposite := sieve[n]
if isComposite {
delete(sieve, n)
m := n + step
for sieve[m] != 0 {
m += step
}
sieve[m] = step
} else if n < q {
out <- n
} else {
step = p + p
m := n + step
for sieve[m] != 0 {
m += step
}
sieve[m] = step
p = <-primes
q = p * p
}
}
}
```
The output:
```txt
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
```
## Groovy
This solution uses a BitSet for compactness and speed, but in [[Groovy]], BitSet has full List semantics. It also uses both the "square root of the boundary" shortcut and the "square of the prime" shortcut.
```groovy
def sievePrimes = { bound ->
def isPrime = new BitSet(bound)
isPrime[0..1] = false
isPrime[2..bound] = true
(2..(Math.sqrt(bound))).each { pc ->
if (isPrime[pc]) {
((pc**2)..bound).step(pc) { isPrime[it] = false }
}
}
(0..bound).findAll { isPrime[it] }
}
```
Test:
```groovy
println sievePrimes(100)
```
Output:
```txt
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]
```
=={{header|GW-BASIC}}==
```qbasic
10 INPUT "ENTER NUMBER TO SEARCH TO: ";LIMIT
20 DIM FLAGS(LIMIT)
30 FOR N = 2 TO SQR (LIMIT)
40 IF FLAGS(N) < > 0 GOTO 80
50 FOR K = N * N TO LIMIT STEP N
60 FLAGS(K) = 1
70 NEXT K
80 NEXT N
90 REM DISPLAY THE PRIMES
100 FOR N = 2 TO LIMIT
110 IF FLAGS(N) = 0 THEN PRINT N;", ";
120 NEXT N
```
## Haskell
===Mutable unboxed arrays, odds only===
Mutable array of unboxed Bools indexed by Ints, representing odds only:
```haskell
import Control.Monad (forM_, when)
import Control.Monad.ST
import Data.Array.ST
import Data.Array.Unboxed
sieveUO :: Int -> UArray Int Bool
sieveUO top = runSTUArray $ do
let m = (top-1) `div` 2
r = floor . sqrt $ fromIntegral top + 1
sieve <- newArray (1,m) True -- :: ST s (STUArray s Int Bool)
forM_ [1..r `div` 2] $ \i -> do -- prime(i) = 2i+1
isPrime <- readArray sieve i -- ((2i+1)^2-1)`div`2 = 2i(i+1)
when isPrime $ do
forM_ [2*i*(i+1), 2*i*(i+2)+1..m] $ \j -> do
writeArray sieve j False
return sieve
primesToUO :: Int -> [Int]
primesToUO top | top > 1 = 2 : [2*i + 1 | (i,True) <- assocs $ sieveUO top]
| otherwise = []
```
This represents ''odds only'' in the array. [http://ideone.com/KwZNc Empirical orders of growth] is ~ n1.2 in ''n'' primes produced, and improving for bigger ''n'' s. Memory consumption is low (array seems to be packed) and growing about linearly with ''n''. Can further be [http://ideone.com/j24jxV significantly sped up] by re-writing the forM_ loops with direct recursion, and using unsafeRead and unsafeWrite operations.
### Immutable arrays
Monolithic sieving array. ''Even'' numbers above 2 are pre-marked as composite, and sieving is done only by ''odd'' multiples of ''odd'' primes:
```haskell
import Data.Array.Unboxed
primesToA m = sieve 3 (array (3,m) [(i,odd i) | i<-[3..m]] :: UArray Int Bool)
where
sieve p a
| p*p > m = 2 : [i | (i,True) <- assocs a]
| a!p = sieve (p+2) $ a//[(i,False) | i <- [p*p, p*p+2*p..m]]
| otherwise = sieve (p+2) a
```
Its performance sharply depends on compiler optimizations. Compiled with -O2 flag in the presence of the explicit type signature, it is very fast in producing first few million primes. (//) is an array update operator.
===Immutable arrays, by segments===
Works by segments between consecutive primes' squares. Should be the fastest of non-monadic code. ''Evens'' are entirely ignored:
```haskell
import Data.Array.Unboxed
primesSA = 2 : prs ()
where
prs () = 3 : sieve 3 [] (prs ())
sieve x fs (p:ps) = [i*2 + x | (i,True) <- assocs a]
++ sieve (p*p) fs2 ps
where
q = (p*p-x)`div`2
fs2 = (p,0) : [(s, rem (y-q) s) | (s,y) <- fs]
a :: UArray Int Bool
a = accumArray (\ b c -> False) True (1,q-1)
[(i,()) | (s,y) <- fs, i <- [y+s, y+s+s..q]]
```
===Basic list-based sieve===
Straightforward implementation of the sieve of Eratosthenes in its original bounded form. This finds primes in gaps between the composites, and composites as an enumeration of each prime's multiples.
```haskell
primesTo m = eratos [2..m] where
eratos (p : xs)
| p*p > m = p : xs
| otherwise = p : eratos (xs `minus` [p*p, p*p+p..m])
-- map (p*) [p..]
-- map (p*) (p:xs) -- (Euler's sieve)
minus a@(x:xs) b@(y:ys) = case compare x y of
LT -> x : minus xs b
EQ -> minus xs ys
GT -> minus a ys
minus a b = a
```
Its time complexity is similar to that of optimal [[Primality_by_trial_division#Haskell|trial division]] because of limitations of Haskell linked lists, where (minus a b) takes time proportional to length(union a b) and not (length b), as achieved in imperative setting with direct-access memory. Uses ordered list representation of sets.
This is reasonably useful up to ranges of fifteen million or about the first million primes.
### Unbounded list based sieve
Unbounded, "naive", too eager to subtract (see above for the definition of minus):
```haskell
primesE = sieve [2..]
where
sieve (p:xs) = p : sieve (minus xs [p, p+p..])
-- unfoldr (\(p:xs)-> Just (p, minus xs [p, p+p..])) [2..]
```
This is slow, with complexity increasing as a square law or worse so that it is only moderately useful for the first few thousand primes or so.
The number of active streams can be limited to what's strictly necessary by postponement until the square of a prime is seen, getting a massive complexity improvement to better than ~ n1.5 so it can get first million primes or so in a tolerable time:
```haskell
primesPE = 2 : sieve [3..] 4 primesPE
where
sieve (x:xs) q (p:t)
| x < q = x : sieve xs q (p:t)
| otherwise = sieve (minus xs [q, q+p..]) (head t^2) t
-- fix $ (2:) . concat
-- . unfoldr (\(p:ps,xs)-> Just . second ((ps,) . (`minus` [p*p, p*p+p..]))
-- . span (< p*p) $ xs) . (,[3..])
```
Transposing the workflow, going by segments between the consecutive squares of primes:
```haskell
import Data.List (inits)
primesSE = 2 : sieve 3 4 (tail primesSE) (inits primesSE)
where
sieve x q ps (fs:ft) =
foldl minus [x..q-1] [[n, n+f..q-1] | f <- fs, let n=div x f * f]
-- [i|(i,True) <- assocs ( accumArray (\ b c -> False)
-- True (x,q-1) [(i,()) | f <- fs, let n=div(x+f-1)f*f,
-- i <- [n, n+f..q-1]] :: UArray Int Bool )]
++ sieve q (head ps^2) (tail ps) ft
```
The basic gradually-deepening left-leaning (((a-b)-c)- ... ) workflow of foldl minus a bs above can be rearranged into the right-leaning (a-(b+(c+ ... ))) workflow of minus a (foldr union [] bs). This is the idea behind Richard Bird's unbounded code presented in [http://www.cs.hmc.edu/~oneill/papers/Sieve-JFP.pdf M. O'Neill's article], equivalent to:
```haskell
primesB = _Y ( (2:) . minus [3..] . foldr (\p-> (p*p :) . union [p*p+p, p*p+2*p..]) [] )
-- = _Y ( (2:) . minus [3..] . _LU . map(\p-> [p*p, p*p+p..]) )
-- _LU ((x:xs):t) = x : (union xs . _LU) t -- linear folding big union
_Y g = g (_Y g) -- = g (g (g ( ... ))) non-sharing multistage fixpoint combinator
-- = g . g . g . ... ... = g^inf
-- = let x = g x in g x -- = g (fix g) two-stage fixpoint combinator
-- = let x = g x in x -- = fix g sharing fixpoint combinator
union a@(x:xs) b@(y:ys) = case compare x y of
LT -> x : union xs b
EQ -> x : union xs ys
GT -> y : union a ys
```
Using _Y is meant to guarantee the separate supply of primes to be independently calculated, recursively, instead of the same one being reused, corecursively; thus the memory footprint is drastically reduced. This idea was introduced by M. ONeill as a double-staged production, with a separate primes feed.
The above code is also useful to a range of the first million primes or so. The code can be further optimized by fusing minus [3..] into one function, preventing a space leak with the newer GHC versions, getting the function gaps defined below.
====Tree-merging incremental sieve====
Linear merging structure can further be replaced with an wiki.haskell.org/Prime_numbers#Tree_merging indefinitely deepening to the right tree-like structure, (a-(b+((c+d)+( ((e+f)+(g+h)) + ... )))).
This merges primes' multiples streams in a ''tree''-like fashion, as a sequence of balanced trees of union nodes, likely achieving theoretical time complexity only a ''log n'' factor above the optimal ''n log n log (log n)'', for ''n'' primes produced. Indeed, empirically it runs at about ''~ n1.2'' (for producing first few million primes), similarly to priority-queue–based version of M. O'Neill's, and with very low space complexity too (not counting the produced sequence of course):
```haskell
primes :: [Int]
primes = 2 : _Y ( (3:) . gaps 5 . _U . map(\p-> [p*p, p*p+2*p..]) )
gaps k s@(c:cs) | k < c = k : gaps (k+2) s -- ~= ([k,k+2..] \\ s)
| otherwise = gaps (k+2) cs -- when null(s\\[k,k+2..])
_U ((x:xs):t) = x : (union xs . _U . pairs) t -- tree-shaped folding big union
where -- ~= nub . sort . concat
pairs (xs:ys:t) = union xs ys : pairs t
```
Works with odds only, the simplest kind of wheel. Here's the [http://ideone.com/qpnqe test entry] on Ideone.com, and a [http://ideone.com/p0e81 comparison with more versions].
### =With Wheel=
Using _U defined above,
```haskell
primesW :: [Int]
primesW = [2,3,5,7] ++ _Y ( (11:) . gapsW 13 (tail wheel) . _U .
map (\p->
map (p*) . dropWhile (< p) $
scanl (+) (p - rem (p-11) 210) wheel) )
gapsW k (d:w) s@(c:cs) | k < c = k : gapsW (k+d) w s -- set difference
| otherwise = gapsW (k+d) w cs -- k==c
wheel = 2:4:2:4:6:2:6:4:2:4:6:6:2:6:4:2:6:4:6:8:4:2:4:2: -- gaps = (`gapsW` cycle [2])
4:8:6:4:6:2:4:6:2:6:6:4:2:4:6:2:6:4:2:4:2:10:2:10:wheel
-- cycle $ zipWith (-) =<< tail $ [i | i <- [11..221], gcd i 210 == 1]
```
Used [[Emirp_primes#List-based|here]] and [[Extensible_prime_generator#List_based|here]].
### Priority Queue based incremental sieve
The above work is derived from the Epilogue of the Melissa E. O'Neill paper which is much referenced with respect to incremental functional sieves; however, that paper is now dated and her comments comparing list based sieves to her original work leading up to a Priority Queue based implementation is no longer current given more recent work such as the above Tree Merging version. Accordingly, a modern "odd's-only" Priority Queue version is developed here for more current comparisons between the above list based incremental sieves and a continuation of O'Neill's work.
In order to implement a Priority Queue version with Haskell, an efficient Priority Queue, which is not part of the standard Haskell libraries is required. A Min Heap implementation is likely best suited for this task in providing the most efficient frequently used peeks of the next item in the queue and replacement of the first item in the queue (not using a "pop" followed by a "push) with "pop" operations then not used at all, and "push" operations used relatively infrequently. Judging by O'Neill's use of an efficient "deleteMinAndInsert" operation which she states "(We provide deleteMinAndInsert becausea heap-based implementation can support this operation with considerably less rearrangement than a deleteMin followed by an insert.)", which statement is true for a Min Heap Priority Queue and not others, and her reference to a priority queue by (Paulson, 1996), the queue she used is likely the one as provided as a simple true functional [http://rosettacode.org/wiki/Priority_queue#Haskell Min Heap implementation on RosettaCode], from which the essential functions are reproduced here:
```haskell
data PriorityQ k v = Mt
| Br !k v !(PriorityQ k v) !(PriorityQ k v)
deriving (Eq, Ord, Read, Show)
emptyPQ :: PriorityQ k v
emptyPQ = Mt
peekMinPQ :: PriorityQ k v -> Maybe (k, v)
peekMinPQ Mt = Nothing
peekMinPQ (Br k v _ _) = Just (k, v)
pushPQ :: Ord k => k -> v -> PriorityQ k v -> PriorityQ k v
pushPQ wk wv Mt = Br wk wv Mt Mt
pushPQ wk wv (Br vk vv pl pr)
| wk <= vk = Br wk wv (pushPQ vk vv pr) pl
| otherwise = Br vk vv (pushPQ wk wv pr) pl
siftdown :: Ord k => k -> v -> PriorityQ k v -> PriorityQ k v -> PriorityQ k v
siftdown wk wv Mt _ = Br wk wv Mt Mt
siftdown wk wv (pl @ (Br vk vv _ _)) Mt
| wk <= vk = Br wk wv pl Mt
| otherwise = Br vk vv (Br wk wv Mt Mt) Mt
siftdown wk wv (pl @ (Br vkl vvl pll plr)) (pr @ (Br vkr vvr prl prr))
| wk <= vkl && wk <= vkr = Br wk wv pl pr
| vkl <= vkr = Br vkl vvl (siftdown wk wv pll plr) pr
| otherwise = Br vkr vvr pl (siftdown wk wv prl prr)
replaceMinPQ :: Ord k => k -> v -> PriorityQ k v -> PriorityQ k v
replaceMinPQ wk wv Mt = Mt
replaceMinPQ wk wv (Br _ _ pl pr) = siftdown wk wv pl pr
```
The "peekMin" function retrieves both of the key and value in a tuple so processing is required to access whichever is required for further processing. As well, the output of the peekMin function is a Maybe with the case of an empty queue providing a Nothing output.
The following code is O'Neill's original odds-only code (without wheel factorization) from her paper slightly adjusted as per the requirements of this Min Heap implementation as laid out above; note the `seq` adjustments to the "adjust" function to make the evaluation of the entry tuple more strict for better efficiency:
```haskell
-- (c) 2006-2007 Melissa O'Neill. Code may be used freely so long as
-- this copyright message is retained and changed versions of the file
-- are clearly marked.
-- the only changes are the names of the called PQ functions and the
-- included processing for the result of the peek function being a maybe tuple.
primesPQ() = 2 : sieve [3,5..]
where
sieve [] = []
sieve (x:xs) = x : sieve' xs (insertprime x xs emptyPQ)
where
insertprime p xs table = pushPQ (p*p) (map (* p) xs) table
sieve' [] table = []
sieve' (x:xs) table
| nextComposite <= x = sieve' xs (adjust table)
| otherwise = x : sieve' xs (insertprime x xs table)
where
nextComposite = case peekMinPQ table of
Just (c, _) -> c
adjust table
| n <= x = adjust (replaceMinPQ n' ns table)
| otherwise = table
where (n, n':ns) = case peekMinPQ table of
Just tpl -> tpl
```
The above code is almost four times slower than the version of the Tree Merging sieve above for the first million primes although it is about the same speed as the original Richard Bird sieve with the "odds-only" adaptation as above. It is slow and uses a huge amount of memory for primarily one reason: over eagerness in adding prime composite streams to the queue, which are added as the primes are listed rather than when they are required as the output primes stream reaches the square of a given base prime; this over eagerness also means that the processed numbers must have a large range in order to not overflow when squared (as in the default Integer = infinite precision integers as used here and by O'Neill, but Int64's or Word64's would give a practical range) which processing of wide range numbers adds processing and memory requirement overhead. Although O'Neill's code is elegant, it also loses some efficiency due to the extensive use of lazy list processing, not all of which is required even for a wheel factorization implementation.
The following code is adjusted to reduce the amount of lazy list processing and to add a secondary base primes stream (or a succession of streams when the combinator is used) so as to overcome the above problems and reduce memory consumption to only that required for the primes below the square root of the currently sieved number; using this means that 32-bit Int's are sufficient for a reasonable range and memory requirements become relatively negligible:
```haskell
primesPQx :: () -> [Int]
primesPQx() = 2 : _Y ((3 :) . sieve 5 emptyPQ 9) -- initBasePrms
where
_Y g = g (_Y g) -- non-sharing multi-stage fixpoint combinator OR
-- initBasePrms = 3 : sieve 5 emptyPQ 9 initBasePrms -- single stage
insertprime p table = let adv = 2 * p in let nv = p * p + adv in
nv `seq` pushPQ nv adv table
sieve n table q bps@(bp:bps')
| n >= q = let nbp = head bps' in
sieve (n + 2) (insertprime bp table) (nbp * nbp) bps'
| n >= nextComposite = sieve (n + 2) (adjust table) q bps
| otherwise = n : sieve (n + 2) table q bps
where
nextComposite = case peekMinPQ table of
Nothing -> q -- at beginning when queue empty
Just (c, _) -> c
adjust table
| c <= n = let nc = c + adv in
nc `seq` adjust (replaceMinPQ nc adv table)
| otherwise = table
where (c, adv) = case peekMinPQ table of
Just ct -> ct
```
The above code is over five times faster than the previous (O'Neill) Priority Queue code and about half again faster than the Tree Merging code for a range of a million primes, and will always be faster as the Min Heap is slightly more efficient than Tree Merging due to better tree balancing.
All of these codes including the list based ones would enjoy about the same constant factor improvement of up to about four times the speed with the application of maximum wheel factorization.
### Page Segmented Sieve using a mutable unboxed array
All of the above unbounded sieves are quite limited in practical sieving range due to the large constant factor overheads in computation, making them mostly just interesting intellectual exercises other than for small ranges of about the first million to ten million primes; the following '''"odds-only''' page-segmented version using (bit-packed internally) mutable unboxed arrays is about 50 times faster than the fastest of the above algorithms for ranges of about that and higher, making it practical for the first several hundred million primes:
```haskell
import Data.Bits
import Data.Array.Base
import Control.Monad.ST
import Data.Array.ST (runSTUArray, STUArray(..))
type PrimeType = Int
szPGBTS = (2^14) * 8 :: PrimeType -- CPU L1 cache in bits
primesPaged :: () -> [PrimeType]
primesPaged() = 2 : _Y (listPagePrms . pagesFrom 0) where
_Y g = g (_Y g) -- non-sharing multi-stage fixpoint combinator
listPagePrms (hdpg @ (UArray lowi _ rng _) : tlpgs) =
let loop i = if i >= rng then listPagePrms tlpgs
else if unsafeAt hdpg i then loop (i + 1)
else let ii = lowi + fromIntegral i in
case 3 + ii + ii of
p -> p `seq` p : loop (i + 1) in loop 0
makePg lowi bps = runSTUArray $ do
let limi = lowi + szPGBTS - 1
let nxt = 3 + limi + limi -- last candidate in range
cmpsts <- newArray (lowi, limi) False
let pbts = fromIntegral szPGBTS
let cull (p:ps) =
let sqr = p * p in
if sqr > nxt then return cmpsts
else let pi = fromIntegral p in
let cullp c = if c > pbts then return ()
else do
unsafeWrite cmpsts c True
cullp (c + pi) in
let a = (sqr - 3) `shiftR` 1 in
let s = if a >= lowi then fromIntegral (a - lowi)
else let r = fromIntegral ((lowi - a) `rem` p) in
if r == 0 then 0 else pi - r in
do { cullp s; cull ps}
if lowi == 0 then do
pg0 <- unsafeFreezeSTUArray cmpsts
cull $ listPagePrms [pg0]
else cull bps
pagesFrom lowi bps =
let cf lwi = case makePg lwi bps of
pg -> pg `seq` pg : cf (lwi + szPGBTS) in cf lowi
```
The above code is currently implemented to use "Int" as the prime type but one can change the "PrimeType" to "Int64" (importing Data.Int) or "Word64" (importing Data.Word) to extend the range to its maximum practical range of above 10^14 or so. Note that for larger ranges that one will want to set the "szPGBTS" to something close to the CPU L2 or even L3 cache size (up to 8 Megabytes = 2^23 for an Intel i7) for a slight cost in speed (about a factor of 1.5) but so that it still computes fairly efficiently as to memory access up to those large ranges. It would be quite easy to modify the above code to make the page array size automatically increase in size with increasing range.
The above code takes only a few tens of milliseconds to compute the first million primes and a few seconds to calculate the first 50 million primes, with over half of those times expended in just enumerating the result lazy list, with even worse times when using 64-bit list processing (especially with 32-bit versions of GHC). A further improvement to reduce the computational cost of repeated list processing across the base pages for every page segment would be to store the required base primes (or base prime gaps) in an array that gets extended in size by factors of two (by copying the old array to the new extended array) as the number of base primes increases; in that way the scans across base primes per page segment would just be array accesses which are much faster than list enumeration.
Unlike many other other unbounded examples, this algorithm has the true Sieve of Eratosthenes computational time complexity of O(n log log n) where n is the sieving range with no extra "log n" factor while having a very low computational time cost per composite number cull of less than ten CPU clock cycles per cull (well under as in under 4 clock cycles for the Intel i7 using a page buffer size of the CPU L1 cache size).
There are other ways to make the algorithm faster including high degrees of wheel factorization, which can reduce the number of composite culling operations by a factor of about four for practical ranges, and multi-processing which can reduce the computation time proportionally to the number of available independent CPU cores, but there is little point to these optimizations as long as the lazy list enumeration is the bottleneck as it is starting to be in the above code. To take advantage of those optimizations, functions need to be provided that can compute the desired results without using list processing.
For ranges above about 10^14 where culling spans begin to exceed even an expanded size page array, other techniques need to be adapted such as the use of a "bucket sieve" which tracks the next page that larger base prime culling sequences will "hit" to avoid redundant (and time expensive) start address calculations for base primes that don't "hit" the current page.
However, even with the above code and its limitations for large sieving ranges, the speeds will never come close to as slow as the other "incremental" sieve algorithms, as the time will never exceed about 100 CPU clock cycles per composite number cull, where the fastest of those other algorithms takes many hundreds of CPU clock cycles per cull.
===APL-style===
Rolling set subtraction over the rolling element-wise addition on integers. Basic, slow, worse than quadratic in the number of primes produced, empirically:
```haskell
zipWith (flip (!!)) [0..] -- or: take n . last . take n ...
. scanl1 minus
. scanl1 (zipWith (+)) $ repeat [2..]
```
Or, a wee bit faster:
```haskell
unfoldr (\(a:b:t) -> Just . (head &&& (:t) . (`minus` b)
. tail) $ a)
. scanl1 (zipWith (+)) $ repeat [2..]
```
A bit optimized, much faster, with better complexity,
```haskell
tail . concat
. unfoldr (\(a:b:t) -> Just . second ((:t) . (`minus` b))
. span (< head b) $ a)
. scanl1 (zipWith (+) . tail) $ tails [1..]
-- $ [ [n*n, n*n+n..] | n <- [1..] ]
```
getting nearer to the functional equivalent of the primesPE above, i.e.
```haskell
fix ( (2:) . concat
. unfoldr (\(a:b:t) -> Just . second ((:t) . (`minus` b))
. span (< head b) $ a)
. ([3..] :) . map (\p-> [p*p, p*p+p..]) )
```
An illustration:
```haskell>
mapM_ (print . take 15) $ take 10 $ scanl1 (zipWith(+)) $ repeat [2..]
[ 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]
[ 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32]
[ 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48]
[ 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64]
[ 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80]
[ 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96]
[ 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98,105,112]
[ 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96,104,112,120,128]
[ 18, 27, 36, 45, 54, 63, 72, 81, 90, 99,108,117,126,135,144]
[ 20, 30, 40, 50, 60, 70, 80, 90,100,110,120,130,140,150,160]
> mapM_ (print . take 15) $ take 10 $ scanl1 (zipWith(+) . tail) $ tails [1..]
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]
[ 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32]
[ 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51]
[ 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72]
[ 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95]
[ 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96,102,108,114,120]
[ 49, 56, 63, 70, 77, 84, 91, 98,105,112,119,126,133,140,147]
[ 64, 72, 80, 88, 96,104,112,120,128,136,144,152,160,168,176]
[ 81, 90, 99,108,117,126,135,144,153,162,171,180,189,198,207]
[100,110,120,130,140,150,160,170,180,190,200,210,220,230,240]
```
## HicEst
```hicest
REAL :: N=100, sieve(N)
sieve = $ > 1 ! = 0 1 1 1 1 ...
DO i = 1, N^0.5
IF( sieve(i) ) THEN
DO j = i^2, N, i
sieve(j) = 0
ENDDO
ENDIF
ENDDO
DO i = 1, N
IF( sieve(i) ) WRITE() i
ENDDO
```
=={{header|Icon}} and {{header|Unicon}}==
```Icon
procedure main()
sieve(100)
end
procedure sieve(n)
local p,i,j
p:=list(n, 1)
every i:=2 to sqrt(n) & j:= i+i to n by i & p[i] == 1
do p[j] := 0
every write(i:=2 to n & p[i] == 1 & i)
end
```
Alternatively using sets
```Icon
procedure main()
sieve(100)
end
procedure sieve(n)
primes := set()
every insert(primes,1 to n)
every member(primes,i := 2 to n) do
every delete(primes,i + i to n by i)
delete(primes,1)
every write(!sort(primes))
end
```
## J
This problem is a classic example of how J can be used to represent mathematical concepts.
J uses x|y ([http://www.jsoftware.com/help/dictionary/d230.htm residue]) to represent the operation of finding the remainder during integer division of y divided by x
```J
10|13
3
```
And x|/y gives us a [http://www.jsoftware.com/help/dictionary/d420.htm table] with all possibilities from x and all possibilities from y.
```J
2 3 4 |/ 2 3 4
0 1 0
2 0 1
2 3 0
```
Meanwhile, |/~y ([http://www.jsoftware.com/help/dictionary/d220v.htm reflex]) copies the right argument and uses it as the left argment.
```J
|/~ 0 1 2 3 4
0 1 2 3 4
0 0 0 0 0
0 1 0 1 0
0 1 2 0 1
0 1 2 3 0
```
(Bigger examples might make the patterns more obvious but they also take up more space.)
By the way, we can ask J to count out the first N integers for us using i. ([http://www.jsoftware.com/help/dictionary/didot.htm integers]):
```J
i. 5
0 1 2 3 4
```
Anyways, the 0s in that last table represent the Sieve of Eratosthenes (in a symbolic or mathematical sense), and we can use = ([http://www.jsoftware.com/help/dictionary/d000.htm equal]) to find them.
```J
0=|/~ i.5
1 0 0 0 0
1 1 1 1 1
1 0 1 0 1
1 0 0 1 0
1 0 0 0 1
```
Now all we need to do is add them up, using / ([http://www.jsoftware.com/help/dictionary/d420.htm insert]) in its single argument role to insert + between each row of that last table.
```J
+/0=|/~ i.5
5 1 2 2 3
```
The sieve wants the cases where we have two divisors:
```J
2=+/0=|/~ i.5
0 0 1 1 0
```
And we just want to know the positions of the 1s in that list, which we can find using I. ([http://www.jsoftware.com/help/dictionary/dicapdot.htm indices]):
```J
I.2=+/0=|/~ i.5
2 3
I.2=+/0=|/~ i.100
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
```
And we might want to express this sentence as a definition of a word that lets us use it with an arbitrary argument. There are a variety of ways of doing this. For example:
```J
sieve0=: verb def 'I.2=+/0=|/~ i.y'
```
That said, this fails with an argument of 2 (instead of giving an empty list of the primes smaller than 2, it gives a list with one element: 0). Working through why this is and why this matters can be an informative exercise. But, assuming this matters, we need to add some guard logic to prevent that problem:
```J
sieve0a=: verb def 'I.(y>2)*2=+/0=|/~ i.y'
```
Of course, we can also express this in an even more elaborate fashion. The elaboration makes more efficient use of resources for large arguments, at the expense of less efficient use of resources for small arguments:
```J
sieve1=: 3 : 0
m=. <.%:y
z=. $0
b=. y{.1
while. m>:j=. 1+b i. 0 do.
b=. b+.y$(-j){.1
z=. z,j
end.
z,1+I.-.b
)
```
"Wheels" may be implemented as follows:
```J
sieve2=: 3 : 0
m=. <.%:y
z=. y (>:#]) 2 3 5 7
b=. 1,}.y$+./(*/z)$&>(-z){.&.>1
while. m>:j=. 1+b i. 0 do.
b=. b+.y$(-j){.1
z=. z,j
end.
z,1+I.-.b
)
```
The use of 2 3 5 7 as wheels provides a
20% time improvement for n=1000 and 2% for n=1e6 but note that sieve2 is still 25 times slower than i.&.(p:inv) for n=1e6. Then again, the value of the sieve of eratosthenes was not efficiency but simplicity. So perhaps we should ignore resource consumption issues and instead focus on intermediate results for reasonably sized example problems?
```J
0=|/~ i.8
1 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1
1 0 1 0 1 0 1 0
1 0 0 1 0 0 1 0
1 0 0 0 1 0 0 0
1 0 0 0 0 1 0 0
1 0 0 0 0 0 1 0
1 0 0 0 0 0 0 1
```
Columns with two "1" values correspond to prime numbers.
'''Alternate Implementation'''
If you feel that the intermediate results, above, are not enough "sieve-like" another approach could be:
```J
sieve=:verb define
seq=: 2+i.y-1 NB. 2 thru y
n=. 2
l=. #seq
whilst. -.seq-:prev do.
prev=. seq
mask=. l{.1-(0{.~n-1),1}.l$n{.1
seq=. seq * mask
n=. {.((n-1)}.seq)-.0
end.
seq -. 0
)
```
Example use:
```J
sieve 100
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
```
To see intermediate results, let's show them:
```J
label=:dyad def 'echo x,":y'
sieve=:verb define
'seq ' label seq=: 2+i.y-1 NB. 2 thru y
'n ' label n=. 2
'l ' label l=. #seq
whilst. -.seq-:prev do.
prev=. seq
'mask ' label mask=. l{.1-(0{.~n-1),1}.l$n{.1
'seq ' label seq=. seq * mask
'n ' label n=. {.((n-1)}.seq)-.0
end.
seq -. 0
)
seq 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
n 2
l 59
mask 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
seq 2 3 0 5 0 7 0 9 0 11 0 13 0 15 0 17 0 19 0 21 0 23 0 25 0 27 0 29 0 31 0 33 0 35 0 37 0 39 0 41 0 43 0 45 0 47 0 49 0 51 0 53 0 55 0 57 0 59 0
n 3
mask 1 1 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0
seq 2 3 0 5 0 7 0 0 0 11 0 13 0 0 0 17 0 19 0 0 0 23 0 25 0 0 0 29 0 31 0 0 0 35 0 37 0 0 0 41 0 43 0 0 0 47 0 49 0 0 0 53 0 55 0 0 0 59 0
n 5
mask 1 1 1 1 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0
seq 2 3 0 5 0 7 0 0 0 11 0 13 0 0 0 17 0 19 0 0 0 23 0 0 0 0 0 29 0 31 0 0 0 0 0 37 0 0 0 41 0 43 0 0 0 47 0 49 0 0 0 53 0 0 0 0 0 59 0
n 7
mask 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1
seq 2 3 0 5 0 7 0 0 0 11 0 13 0 0 0 17 0 19 0 0 0 23 0 0 0 0 0 29 0 31 0 0 0 0 0 37 0 0 0 41 0 43 0 0 0 47 0 0 0 0 0 53 0 0 0 0 0 59 0
n 11
mask 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1
seq 2 3 0 5 0 7 0 0 0 11 0 13 0 0 0 17 0 19 0 0 0 23 0 0 0 0 0 29 0 31 0 0 0 0 0 37 0 0 0 41 0 43 0 0 0 47 0 0 0 0 0 53 0 0 0 0 0 59 0
n 13
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
```
Another variation on this theme would be:
```J
sieve=:verb define
seq=: 2+i.y-1 NB. 2 thru y
n=. 1
l=. #seq
whilst. -.seq-:prev do.
prev=. seq
n=. 1+n+1 i.~ * (n-1)}.seq
inds=. (2*n)+n*i.(<.l%n)-1
seq=. 0 inds} seq
end.
seq -. 0
)
```
Intermediate results for this variant are left as an exercise for the reader
## Java
```java5
import java.util.LinkedList;
public class Sieve{
public static LinkedList sieve(int n){
if(n < 2) return new LinkedList();
LinkedList primes = new LinkedList();
LinkedList nums = new LinkedList();
for(int i = 2;i <= n;i++){ //unoptimized
nums.add(i);
}
while(nums.size() > 0){
int nextPrime = nums.remove();
for(int i = nextPrime * nextPrime;i <= n;i += nextPrime){
nums.removeFirstOccurrence(i);
}
primes.add(nextPrime);
}
return primes;
}
}
```
To optimize by testing only odd numbers, replace the loop marked "unoptimized" with these lines:
```java5
nums.add(2);
for(int i = 3;i <= n;i += 2){
nums.add(i);
}
```
Version using a BitSet:
```java5
import java.util.LinkedList;
import java.util.BitSet;
public class Sieve{
public static LinkedList sieve(int n){
LinkedList primes = new LinkedList();
BitSet nonPrimes = new BitSet(n+1);
for (int p = 2; p <= n ; p = nonPrimes.nextClearBit(p+1)) {
for (int i = p * p; i <= n; i += p)
nonPrimes.set(i);
primes.add(p);
}
return primes;
}
}
```
### Infinite iterator
An iterator that will generate primes indefinitely (perhaps until it runs out of memory), but very slowly.
```java5
import java.util.Iterator;
import java.util.PriorityQueue;
import java.math.BigInteger;
// generates all prime numbers
public class InfiniteSieve implements Iterator {
private static class NonPrimeSequence implements Comparable {
BigInteger currentMultiple;
BigInteger prime;
public NonPrimeSequence(BigInteger p) {
prime = p;
currentMultiple = p.multiply(p); // start at square of prime
}
@Override public int compareTo(NonPrimeSequence other) {
// sorted by value of current multiple
return currentMultiple.compareTo(other.currentMultiple);
}
}
private BigInteger i = BigInteger.valueOf(2);
// priority queue of the sequences of non-primes
// the priority queue allows us to get the "next" non-prime quickly
final PriorityQueue nonprimes = new PriorityQueue();
@Override public boolean hasNext() { return true; }
@Override public BigInteger next() {
// skip non-prime numbers
for ( ; !nonprimes.isEmpty() && i.equals(nonprimes.peek().currentMultiple); i = i.add(BigInteger.ONE)) {
// for each sequence that generates this number,
// have it go to the next number (simply add the prime)
// and re-position it in the priority queue
while (nonprimes.peek().currentMultiple.equals(i)) {
NonPrimeSequence x = nonprimes.poll();
x.currentMultiple = x.currentMultiple.add(x.prime);
nonprimes.offer(x);
}
}
// prime
// insert a NonPrimeSequence object into the priority queue
nonprimes.offer(new NonPrimeSequence(i));
BigInteger result = i;
i = i.add(BigInteger.ONE);
return result;
}
public static void main(String[] args) {
Iterator sieve = new InfiniteSieve();
for (int i = 0; i < 25; i++) {
System.out.println(sieve.next());
}
}
}
```
```txt
2
3
5
7
11
13
17
19
```
===Infinite iterator with a faster algorithm (sieves odds-only)===
The adding of each discovered prime's incremental step information to the mapping should be postponed until the candidate number reaches the primes square, as it is useless before that point. This drastically reduces the space complexity from O(n/log(n)) to O(sqrt(n/log(n))), in n primes produced, and also lowers the run time complexity due to the use of the hash table based HashMap, which is much more efficient for large ranges.
```java5
import java.util.Iterator;
import java.util.HashMap;
// generates all prime numbers up to about 10 ^ 19 if one can wait 1000's of years or so...
public class SoEInfHashMap implements Iterator {
long candidate = 2;
Iterator baseprimes = null;
long basep = 3;
long basepsqr = 9;
// HashMap of the sequences of non-primes
// the hash map allows us to get the "next" non-prime reasonably quickly
// but further allows re-insertions to take amortized constant time
final HashMap nonprimes = new HashMap<>();
@Override public boolean hasNext() { return true; }
@Override public Long next() {
// do the initial primes separately to initialize the base primes sequence
if (this.candidate <= 5L) if (this.candidate++ == 2L) return 2L; else {
this.candidate++; if (this.candidate == 5L) return 3L; else {
this.baseprimes = new SoEInfHashMap();
this.baseprimes.next(); this.baseprimes.next(); // throw away 2 and 3
return 5L;
} }
// skip non-prime numbers including squares of next base prime
for ( ; this.candidate >= this.basepsqr || //equals nextbase squared => not prime
nonprimes.containsKey(this.candidate); candidate += 2) {
// insert a square root prime sequence into hash map if to limit
if (candidate >= basepsqr) { // if square of base prime, always equal
long adv = this.basep << 1;
nonprimes.put(this.basep * this.basep + adv, adv);
this.basep = this.baseprimes.next();
this.basepsqr = this.basep * this.basep;
}
// else for each sequence that generates this number,
// have it go to the next number (simply add the advance)
// and re-position it in the hash map at an emply slot
else {
long adv = nonprimes.remove(this.candidate);
long nxt = this.candidate + adv;
while (this.nonprimes.containsKey(nxt)) nxt += adv; //unique keys
this.nonprimes.put(nxt, adv);
}
}
// prime
long tmp = candidate; this.candidate += 2; return tmp;
}
public static void main(String[] args) {
int n = 100000000;
long strt = System.currentTimeMillis();
SoEInfHashMap sieve = new SoEInfHashMap();
int count = 0;
while (sieve.next() <= n) count++;
long elpsd = System.currentTimeMillis() - strt;
System.out.println("Found " + count + " primes up to " + n + " in " + elpsd + " milliseconds.");
}
}
```
```txt
Found 5761455 primes up to 100000000 in 4297 milliseconds.
```
===Infinite iterator with a very fast page segmentation algorithm (sieves odds-only)===
Although somewhat faster than the previous infinite iterator version, the above code is still over 10 times slower than an infinite iterator based on array paged segmentation as in the following code, where the time to enumerate/iterate over the found primes (common to all the iterators) is now about half of the total execution time:
```java5
import java.util.Iterator;
import java.util.ArrayList;
// generates all prime numbers up to about 10 ^ 19 if one can wait 100's of years or so...
// practical range is about 10^14 in a week or so...
public class SoEPagedOdds implements Iterator {
private final int BFSZ = 1 << 16;
private final int BFBTS = BFSZ * 32;
private final int BFRNG = BFBTS * 2;
private long bi = -1;
private long lowi = 0;
private final ArrayList bpa = new ArrayList<>();
private Iterator bps;
private final int[] buf = new int[BFSZ];
@Override public boolean hasNext() { return true; }
@Override public Long next() {
if (this.bi < 1) {
if (this.bi < 0) {
this.bi = 0;
return 2L;
}
//this.bi muxt be 0
long nxt = 3 + (this.lowi << 1) + BFRNG;
if (this.lowi <= 0) { // special culling for first page as no base primes yet:
for (int i = 0, p = 3, sqr = 9; sqr < nxt; i++, p += 2, sqr = p * p)
if ((this.buf[i >>> 5] & (1 << (i & 31))) == 0)
for (int j = (sqr - 3) >> 1; j < BFBTS; j += p)
this.buf[j >>> 5] |= 1 << (j & 31);
}
else { // after the first page:
for (int i = 0; i < this.buf.length; i++)
this.buf[i] = 0; // clear the sieve buffer
if (this.bpa.isEmpty()) { // if this is the first page after the zero one:
this.bps = new SoEPagedOdds(); // initialize separate base primes stream:
this.bps.next(); // advance past the only even prime of two
this.bpa.add(this.bps.next().intValue()); // get the next prime (3 in this case)
}
// get enough base primes for the page range...
for (long p = this.bpa.get(this.bpa.size() - 1), sqr = p * p; sqr < nxt;
p = this.bps.next(), this.bpa.add((int)p), sqr = p * p) ;
for (int i = 0; i < this.bpa.size() - 1; i++) {
long p = this.bpa.get(i);
long s = (p * p - 3) >>> 1;
if (s >= this.lowi) // adjust start index based on page lower limit...
s -= this.lowi;
else {
long r = (this.lowi - s) % p;
s = (r != 0) ? p - r : 0;
}
for (int j = (int)s; j < BFBTS; j += p)
this.buf[j >>> 5] |= 1 << (j & 31);
}
}
}
while ((this.bi < BFBTS) &&
((this.buf[(int)this.bi >>> 5] & (1 << ((int)this.bi & 31))) != 0))
this.bi++; // find next marker still with prime status
if (this.bi < BFBTS) // within buffer: output computed prime
return 3 + ((this.lowi + this.bi++) << 1);
else { // beyond buffer range: advance buffer
this.bi = 0;
this.lowi += BFBTS;
return this.next(); // and recursively loop
}
}
public static void main(String[] args) {
long n = 1000000000;
long strt = System.currentTimeMillis();
Iterator gen = new SoEPagedOdds();
int count = 0;
while (gen.next() <= n) count++;
long elpsd = System.currentTimeMillis() - strt;
System.out.println("Found " + count + " primes up to " + n + " in " + elpsd + " milliseconds.");
}
}
```
```txt
Found 50847534 primes up to 1000000000 in 3201 milliseconds.
```
## JavaScript
```javascript
function eratosthenes(limit) {
var primes = [];
if (limit >= 2) {
var sqrtlmt = Math.sqrt(limit) - 2;
var nums = new Array(); // start with an empty Array...
for (var i = 2; i <= limit; i++) // and
nums.push(i); // only initialize the Array once...
for (var i = 0; i <= sqrtlmt; i++) {
var p = nums[i]
if (p)
for (var j = p * p - 2; j < nums.length; j += p)
nums[j] = 0;
}
for (var i = 0; i < nums.length; i++) {
var p = nums[i];
if (p)
primes.push(p);
}
}
return primes;
}
var primes = eratosthenes(100);
if (typeof print == "undefined")
print = (typeof WScript != "undefined") ? WScript.Echo : alert;
print(primes);
```
outputs:
```txt
2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97
```
Substituting the following code for the function for '''an odds-only algorithm using bit packing''' for the array produces code that is many times faster than the above:
```javascript
function eratosthenes(limit) {
var prms = [];
if (limit >= 2) prms = [2];
if (limit >= 3) {
var sqrtlmt = (Math.sqrt(limit) - 3) >> 1;
var lmt = (limit - 3) >> 1;
var bfsz = (lmt >> 5) + 1
var buf = [];
for (var i = 0; i < bfsz; i++)
buf.push(0);
for (var i = 0; i <= sqrtlmt; i++)
if ((buf[i >> 5] & (1 << (i & 31))) == 0) {
var p = i + i + 3;
for (var j = (p * p - 3) >> 1; j <= lmt; j += p)
buf[j >> 5] |= 1 << (j & 31);
}
for (var i = 0; i <= lmt; i++)
if ((buf[i >> 5] & (1 << (i & 31))) == 0)
prms.push(i + i + 3);
}
return prms;
}
```
While the above code is quite fast especially using an efficient JavaScript engine such as Google Chrome's V8, it isn't as elegant as it could be using the features of the new EcmaScript6 specification when it comes out about the end of 2014 and when JavaScript engines including those of browsers implement that standard in that we might choose to implement an incremental algorithm iterators or generators similar to as implemented in Python or F# (yield). Meanwhile, we can emulate some of those features by using a simulation of an iterator class (which is easier than using a call-back function) for an '''"infinite" generator based on an Object dictionary''' as in the following odds-only code written as a JavaScript class:
```javascript
var SoEIncClass = (function () {
function SoEIncClass() {
this.n = 0;
}
SoEIncClass.prototype.next = function () {
this.n += 2;
if (this.n < 7) { // initialization of sequence to avoid runaway:
if (this.n < 3) { // only even of two:
this.n = 1; // odds from here...
return 2;
}
if (this.n < 5)
return 3;
this.dict = {}; // n must be 5...
this.bps = new SoEIncClass(); // new source of base primes
this.bps.next(); // advance past the even prime of two...
this.p = this.bps.next(); // first odd prime (3 in this case)
this.q = this.p * this.p; // set guard
return 5;
} else { // past initialization:
var s = this.dict[this.n]; // may or may not be defined...
if (!s) { // not defined:
if (this.n < this.q) // haven't reached the guard:
return this.n; // found a prime
else { // n === q => not prime but at guard, so:
var p2 = this.p << 1; // the span odds-only is twice prime
this.dict[this.n + p2] = p2; // add next composite of prime to dict
this.p = this.bps.next();
this.q = this.p * this.p; // get next base prime guard
return this.next(); // not prime so advance...
}
} else { // is a found composite of previous base prime => not prime
delete this.dict[this.n]; // advance to next composite of this prime:
var nxt = this.n + s;
while (this.dict[nxt]) nxt += s; // find unique empty slot in dict
this.dict[nxt] = s; // to put the next composite for this base prime
return this.next(); // not prime so advance...
}
}
};
return SoEIncClass;
})();
```
The above code can be used to find the nth prime (which would require estimating the required range limit using the previous fixed range code) by using the following code:
```javascript
var gen = new SoEIncClass();
for (var i = 1; i < 1000000; i++, gen.next());
var prime = gen.next();
if (typeof print == "undefined")
print = (typeof WScript != "undefined") ? WScript.Echo : alert;
print(prime);
```
to produce the following output (about five seconds using Google Chrome's V8 JavaScript engine):
```txt
15485863
```
The above code is considerably slower than the fixed range code due to the multiple method calls and the use of an object as a dictionary, which (using a hash table as its basis for most implementations) will have about a constant O(1) amortized time per operation but has quite a high constant overhead to convert the numeric indices to strings which are then hashed to be used as table keys for the look-up operations as compared to doing this more directly in implementations such as the Python dict with Python's built-in hashing functions for every supported type.
This can be implemented as '''an "infinite" odds-only generator using page segmentation''' for a considerable speed-up with the alternate JavaScript class code as follows:
```javascript
var SoEPgClass = (function () {
function SoEPgClass() {
this.bi = -1; // constructor resets the enumeration to start...
}
SoEPgClass.prototype.next = function () {
if (this.bi < 1) {
if (this.bi < 0) {
this.bi++;
this.lowi = 0; // other initialization done here...
this.bpa = [];
return 2;
} else { // bi must be zero:
var nxt = 3 + (this.lowi << 1) + 262144;
this.buf = new Array();
for (var i = 0; i < 4096; i++) // faster initialization:
this.buf.push(0);
if (this.lowi <= 0) { // special culling for first page as no base primes yet:
for (var i = 0, p = 3, sqr = 9; sqr < nxt; i++, p += 2, sqr = p * p)
if ((this.buf[i >> 5] & (1 << (i & 31))) === 0)
for (var j = (sqr - 3) >> 1; j < 131072; j += p)
this.buf[j >> 5] |= 1 << (j & 31);
} else { // after the first page:
if (!this.bpa.length) { // if this is the first page after the zero one:
this.bps = new SoEPgClass(); // initialize separate base primes stream:
this.bps.next(); // advance past the only even prime of two
this.bpa.push(this.bps.next()); // get the next prime (3 in this case)
}
// get enough base primes for the page range...
for (var p = this.bpa[this.bpa.length - 1], sqr = p * p; sqr < nxt;
p = this.bps.next(), this.bpa.push(p), sqr = p * p) ;
for (var i = 0; i < this.bpa.length; i++) {
var p = this.bpa[i];
var s = (p * p - 3) >> 1;
if (s >= this.lowi) // adjust start index based on page lower limit...
s -= this.lowi;
else {
var r = (this.lowi - s) % p;
s = (r != 0) ? p - r : 0;
}
for (var j = s; j < 131072; j += p)
this.buf[j >> 5] |= 1 << (j & 31);
}
}
}
}
while (this.bi < 131072 && this.buf[this.bi >> 5] & (1 << (this.bi & 31)))
this.bi++; // find next marker still with prime status
if (this.bi < 131072) // within buffer: output computed prime
return 3 + ((this.lowi + this.bi++) << 1);
else { // beyond buffer range: advance buffer
this.bi = 0;
this.lowi += 131072;
return this.next(); // and recursively loop
}
};
return SoEPgClass;
})();
```
The above code is about fifty times faster (about five seconds to calculate 50 million primes to about a billion on the Google Chrome V8 JavaScript engine) than the above dictionary based code.
The speed for both of these "infinite" solutions will also respond to further wheel factorization techniques, especially for the dictionary based version where any added overhead to deal with the factorization wheel will be negligible compared to the dictionary overhead. The dictionary version would likely speed up about a factor of three or a little more with maximum wheel factorization applied; the page segmented version probably won't gain more than a factor of two and perhaps less due to the overheads of array look-up operations.
## JOVIAL
```JOVIAL
START
FILE MYOUTPUT ... $ ''Insufficient information to complete this declaration''
PROC SIEVEE $
'' define the sieve data structure ''
ARRAY CANDIDATES 1000 B $
FOR I =0,1,999 $
BEGIN
'' everything is potentially prime until proven otherwise ''
CANDIDATES($I$) = 1$
END
'' Neither 1 nor 0 is prime, so flag them off ''
CANDIDATES($0$) = 0$
CANDIDATES($1$) = 0$
'' start the sieve with the integer 0 ''
FOR I = 0$
BEGIN
IF I GE 1000$
GOTO DONE$
'' advance to the next un-crossed out number. ''
'' this number must be a prime ''
NEXTI. IF I LS 1000 AND Candidates($I$) EQ 0 $
BEGIN
I = I + 1 $
GOTO NEXTI $
END
'' insure against running off the end of the data structure ''
IF I LT 1000 $
BEGIN
'' cross out all multiples of the prime, starting with 2*p. ''
FOR J=2 $
FOR K=0 $
BEGIN
K = J * I $
IF K GT 999 $
GOTO ADV $
CANDIDATES($K$) = 0 $
J = J + 1 $
END
'' advance to the next candidate ''
ADV. I = I + 1 $
END
END
'' all uncrossed-out numbers are prime (and only those numbers) ''
'' print all primes ''
DONE. OPEN OUTPUT MYOUTPUT $
FOR I=0,1,999$
BEGIN
IF CANDIDATES($I$) NQ 0$
BEGIN
OUTPUT MYOUTPUT I $
END
END
TERM$
```
## jq
Short and sweet ...
```jq
# Denoting the input by $n, which is assumed to be a positive integer,
# eratosthenes/0 produces an array of primes less than or equal to $n:
def eratosthenes:
# erase(i) sets .[i*j] to false for integral j > 1
def erase(i):
if .[i] then reduce range(2; (1 + length) / i) as $j (.; .[i * $j] = false)
else .
end;
(. + 1) as $n
| (($n|sqrt) / 2) as $s
| [null, null, range(2; $n)]
| reduce (2, 1 + (2 * range(1; $s))) as $i (.; erase($i))
| map(select(.));
```
'''Examples''':
```jq
100 | eratosthenes
```
[2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97]
```jq
1e7 | eratosthenes | length
```
664579
## Julia
Started with 2 already in the array, and then test only for odd numbers and push the prime ones onto the array.
```julia
# Returns an array of positive prime numbers less than or equal to lim
function sieve(lim :: Int)
if lim < 2 return [] end
limi :: Int = (lim - 1) ÷ 2 # calculate the required array size
isprime :: Array{Bool} = trues(limi)
llimi :: Int = (isqrt(lim) - 1) ÷ 2 # and calculate maximum root prime index
result :: Array{Int} = [2] #Initial array
for i in 1:limi
if isprime[i]
p = i + i + 1 # 2i + 1
if i <= llimi
for j = (p*p-1)>>>1:p:limi # quick shift/divide in case LLVM doesn't optimize divide by 2 away
isprime[j] = false
end
end
push!(result, p)
end
end
return result
end
```
Alternate version using findall to get all primes at once in the end
```julia
function sieve(n :: Int)
isprime = trues(n)
isprime[1] = false
for p in 2:n
if isprime[p]
j = p * p
if j > n
return findall(isprime)
else
for k in j:p:n
isprime[k] = false
end
end
end
end
end
```
At about 35 seconds for a range of a billion on my Intel Atom i5-Z8350 CPU at 1.92 GHz (single threaded) or about 70 CPU clock cycles per culling operation, the above examples are two of the very slowest ways to compute the Sieve of Eratosthenes over any kind of a reasonable range due to a couple of factors:
# The output primes are extracted to a result array which takes time (and memory) to construct.
# They use the naive "one huge memory array" method, which has poor memory access speed for larger ranges.
Even though the first uses an odds-only algorithm (not noted in the text as is a requirement of the task) that reduces the number of operations by a factor of about two and a half times, it is not faster than the second, which is not odds-only due to the second being set up to take advantage of the `findall` function to directly output the indices of the remaining true values as the found primes; the second is faster due to the first taking longer to push the found primes singly to the constructed array, whereas internally the second first creates the array to the size of the counted true values and then just fills it.
Also, the first uses more memory than necessary in one byte per `Bool` where using a `BitArray` as in the second reduces this by a factor of eight.
If one is going to "crib" the MatLab algorithm as above, one may as well do it using odds-only as per the MatLab built-in. The following alternate code improves on the "Alternate" example above by making it sieve odds-only and adjusting the result array contents after to suit:
```julia
function sieve2(n :: Int)
ni = (n - 1) ÷ 2
isprime = trues(ni)
for i in 1:ni
if isprime[i]
j = 2i * (i + 1)
if j > ni
m = findall(isprime)
map!((i::Int) -> 2i + 1, m, m)
return pushfirst!(m, 2)
else
p = 2i + 1
while j <= ni
isprime[j] = false
j += p
end
end
end
end
end
```
This takes less about 18.5 seconds or 36 CPU cycles per culling operation to find the primes to a billion, but that is still quite slow compared to what can be done below. Note that the result array needs to be created then copied, created by the findall function, then modified in place by the map! function to transform the indices to primes, and finally copied by the pushfirst! function to add the only even prime of two to the beginning, but these operations are quire fast. However, this still consumes a lot of memory, as in about 64 Megabytes for the sieve buffer and over 400 Megabytes for the result (8-byte Int's for 64 bit execution) to sieve to a billion, and culling the huge culling buffer that doesn't fit the CPU cache sizes is what makes it slow.
### Iterator Output
The creation of an output results array is not necessary if the purpose is just to scan across the resulting primes once, they can be output using an iterator (from a `BitArray`) as in the following odds-only code:
```julia
const Prime = UInt64
struct Primes
rangei :: Int64
primebits :: BitArray{1}
function Primes(n :: Int64)
if n < 3
if n < 2 return new(-1, falses(0)) # no elements
else return new((0, trues(0))) end # n = 2: meaning is 1 element of 2
end
limi :: Int = (n - 1) ÷ 2 # calculate the required array size
isprimes :: BitArray = trues(limi)
@inbounds(
for i in 1:limi
p = i + i + 1
start = (p * p - 1) >>> 1 # shift/divide if LLVM doesn't optimize
if start > limi
return new(limi, isprimes)
end
if isprimes[i]
for j in start:p:limi
isprimes[j] = false
end
end
end)
end
end
Base.eltype(::Type{Primes}) = Prime
function Base.length(P::Primes)::Int64
if P.rangei < 0 return 0 end
return 1 + count(P.primebits)
end
function Base.iterate(P::Primes, state::Int = 0)::
Union{Tuple{Prime, Int}, Nothing}
lmt = P.rangei
if state > lmt return nothing end
if state <= 0 return (UInt64(2), 1) end
let
prmbts = P.primebits
i = state
@inbounds(
while i <= lmt && !prmbts[i] i += 1 end)
if i > lmt return nothing end
return (i + i + 1, i + 1)
end
end
```
for which using the following code:
```julia
function bench()
@time length(Primes(100)) # warm up JIT
# println(@time count(x->true, Primes(1000000000))) # about 1.5 seconds slower counting over iteration
println(@time length(Primes(1000000000)))
end
bench()
```
results in the following output:
```txt
0.000031 seconds (3 allocations: 160 bytes)
12.214533 seconds (4 allocations: 59.605 MiB, 0.42% gc time)
50847534
```
This reduces the CPU cycles per culling cycles to about 24.4, but it's still slow due to using the one largish array. Note that counting each iterated prime takes an additional about one and a half seconds, where if all that is required is the count of primes over a range the specialized length function is much faster.
### Page Segmented Algorithm
For any kind of reasonably large range such as a billion, a page segmented version should be used with the pages sized to the CPU caches for much better memory access times. As well, the following odds-only example uses a custom bit packing algorithm for a further two times speed-up, also reducing the memory allocation delays by reusing the sieve buffers when possible (usually possible):
```julia
const Prime = UInt64
const BasePrime = UInt32
const BasePrimesArray = Array{BasePrime,1}
const SieveBuffer = Array{UInt8,1}
# contains a lazy list of a secondary base primes arrays feed
# NOT thread safe; needs a Mutex gate to make it so...
abstract type BPAS end # stands in for BasePrimesArrays, not defined yet
mutable struct BasePrimesArrays <: BPAS
thunk :: Union{Nothing,Function} # problem with efficiency - untyped function!!!!!!!!!
value :: Union{Nothing,Tuple{BasePrimesArray, BPAS}}
BasePrimesArrays(thunk::Function) = new(thunk)
end
Base.eltype(::Type{BasePrimesArrays}) = BasePrime
Base.IteratorSize(::Type{BasePrimesArrays}) = Base.SizeUnknown() # "infinite"...
function Base.iterate(BPAs::BasePrimesArrays, state::BasePrimesArrays = BPAs)
if state.thunk !== nothing
newvalue :: Union{Nothing,Tuple{BasePrimesArray, BasePrimesArrays}} =
state.thunk() :: Union{Nothing,Tuple{BasePrimesArray
, BasePrimesArrays}}
state.value = newvalue
state.thunk = nothing
return newvalue
end
state.value
end
# count the number of zero bits (primes) in a byte array,
# also works for part arrays/slices, best used as an `@view`...
function countComposites(cmpsts::AbstractArray{UInt8,1})
foldl((a, b) -> a + count_zeros(b), cmpsts; init = 0)
end
# converts an entire sieved array of bytes into an array of UInt32 primes,
# to be used as a source of base primes...
function composites2BasePrimesArray(low::Prime, cmpsts::SieveBuffer)
limiti = length(cmpsts) * 8
len :: Int = countComposites(cmpsts)
rslt :: BasePrimesArray = BasePrimesArray(undef, len)
i :: Int = 0
j :: Int = 1
@inbounds(
while i < limiti
if cmpsts[i >>> 3 + 1] & (1 << (i & 7)) == 0
rslt[j] = low + i + i
j += 1
end
i += 1
end)
rslt
end
# sieving work done, based on low starting value for the given buffer and
# the given lazy list of base prime arrays...
function sieveComposites(low::Prime, buffer::Array{UInt8,1},
bpas::BasePrimesArrays)
lowi :: Int = (low - 3) ÷ 2
len :: Int = length(buffer)
limiti :: Int = len * 8 - 1
nexti :: Int = lowi + limiti
for bpa::BasePrimesArray in bpas
for bp::BasePrime in bpa
bpint :: Int = bp
bpi :: Int = (bpint - 3) >>> 1
starti :: Int = 2 * bpi * (bpi + 3) + 3
starti >= nexti && return
if starti >= lowi starti -= lowi
else
r :: Int = (lowi - starti) % bpint
starti = r == 0 ? 0 : bpint - r
end
lmti :: Int = limiti - 40 * bpint
@inbounds(
if bpint <= (len >>> 2) starti <= lmti
for i in 1:8
if starti > limiti break end
mask = convert(UInt8,1) << (starti & 7)
c = starti >>> 3 + 1
while c <= len
buffer[c] |= mask
c += bpint
end
starti += bpint
end
else
c = starti
while c <= limiti
buffer[c >>> 3 + 1] |= convert(UInt8,1) << (c & 7)
c += bpint
end
end)
end
end
return
end
# starts the secondary base primes feed with minimum size in bits set to 4K...
# thus, for the first buffer primes up to 8293,
# the seeded primes easily cover it as 97 squared is 9409.
function makeBasePrimesArrays() :: BasePrimesArrays
cmpsts :: SieveBuffer = Array{UInt8,1}(undef, 512)
function nextelem(low::Prime, bpas::BasePrimesArrays) ::
Tuple{BasePrimesArray, BasePrimesArrays}
# calculate size so that the bit span is at least as big as the
# maximum culling prime required, rounded up to minsizebits blocks...
reqdsize :: Int = 2 + isqrt(1 + low)
size :: Int = (reqdsize ÷ 4096 + 1) * 4096 ÷ 8 # size in bytes
if size > length(cmpsts) cmpsts = Array{UInt8,1}(undef, size) end
fill!(cmpsts, 0)
sieveComposites(low, cmpsts, bpas)
arr :: BasePrimesArray = composites2BasePrimesArray(low, cmpsts)
next :: Prime = low + length(cmpsts) * 8 * 2
arr, BasePrimesArrays(() -> nextelem(next, bpas))
end
# pre-seeding breaks recursive race,
# as only known base primes used for first page...
preseedarr :: BasePrimesArray = # pre-seed to 100, can sieve to 10,000...
[ 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41
, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
]
nextfunc :: Function = () ->
(nextelem(convert(Prime,101), makeBasePrimesArrays()))
firstfunc :: Function = () -> (preseedarr, BasePrimesArrays(nextfunc))
BasePrimesArrays(firstfunc)
end
# an iterator over successive sieved buffer composite arrays,
# returning a tuple of the value represented by the lowest possible prime
# in the sieved composites array and the array itself;
# the array has a 16 Kilobytes minimum size (CPU L1 cache), but
# will grow so that the bit span is larger than the
# maximum culling base prime required, possibly making it larger than
# the L1 cache for large ranges, but still reasonably efficient using
# the L2 cache: very efficient up to about 16e9 range;
# reasonably efficient to about 2.56e14 for two Megabyte L2 cache = > 1 week...
struct PrimesPages
baseprimes :: BasePrimesArrays
PrimesPages() = new(makeBasePrimesArrays())
end
Base.eltype(::Type{PrimesPages}) = SieveBuffer
Base.IteratorSize(::Type{PrimesPages}) = Base.SizeUnknown() # "infinite"...
function Base.iterate(PP::PrimesPages,
state :: Tuple{Prime,SieveBuffer} =
( convert(Prime,3), Array{UInt8,1}(undef,16384) ))
(low, cmpsts) = state
# calculate size so that the bit span is at least as big as the
# maximum culling prime required, rounded up to minsizebits blocks...
reqdsize :: Int = 2 + isqrt(1 + low)
size :: Int = (reqdsize ÷ 131072 + 1) * 131072 ÷ 8 # size in bytes
if size > length(cmpsts) cmpsts = Array{UInt8,1}(undef, size) end
fill!(cmpsts, 0)
sieveComposites(low, cmpsts, PP.baseprimes)
newlow :: Prime = low + length(cmpsts) * 8 * 2
( low, cmpsts ), ( newlow, cmpsts )
end
function countPrimesTo(range::Prime) :: Int64
range < 3 && ((range < 2 && return 0) || return 1)
count :: Int64 = 1
for ( low, cmpsts ) in PrimesPages() # almost never exits!!!
if low + length(cmpsts) * 8 * 2 > range
lasti :: Int = (range - low) ÷ 2
count += countComposites(@view cmpsts[1:lasti >>> 3])
count += count_zeros(cmpsts[lasti >>> 3 + 1] |
(0xFE << (lasti & 7)))
return count
end
count += countComposites(cmpsts)
end
count
end
# iterator over primes from above page iterator;
# unless doing something special with individual primes, usually unnecessary;
# better to do manipulations based on the composites bit arrays...
# takes at least as long to enumerate the primes as sieve them...
mutable struct PrimesPaged
primespages :: PrimesPages
primespageiter :: Tuple{Tuple{Prime,SieveBuffer},Tuple{Prime,SieveBuffer}}
PrimesPaged() = let PP = PrimesPages(); new(PP, Base.iterate(PP)) end
end
Base.eltype(::Type{PrimesPaged}) = Prime
Base.IteratorSize(::Type{PrimesPaged}) = Base.SizeUnknown() # "infinite"...
function Base.iterate(PP::PrimesPaged, state::Int = -1 )
state < 0 && return Prime(2), 0
(low, cmpsts) = PP.primespageiter[1]
len = length(cmpsts) * 8
@inbounds(
while state < len && cmpsts[state >>> 3 + 1] &
(UInt8(1) << (state & 7)) != 0
state += 1
end)
if state >= len
PP.primespageiter = Base.iterate(PP.primespages, PP.primespageiter[2])
return Base.iterate(PP, 0)
end
low + state + state, state + 1
end
```
When tested with the following code:
```julia
function bench()
print("( ")
for p in PrimesPaged() p > 100 && break; print(p, " ") end
println(")")
countPrimesTo(Prime(100)) # warm up JIT
#=
println(@time let count = 0
for p in PrimesPaged()
p > 1000000000 && break
count += 1
end; count end) # much slower counting over iteration
=#
println(@time countPrimesTo(Prime(1000000000)))
end
bench()
```
it produces the following:
```txt
( 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 )
1.947145 seconds (59 allocations: 39.078 KiB)
50847534
```
Note that "the slow way" as commented out in the code takes an extra about 4.85 seconds to count the primes to a billion, or longer to enumerate the primes than to cull the composites; this makes further work in making this yet faster pointless unless techniques such as the one used here to count the number of found primes by just counting the un-cancelled bit representations in the sieved sieve buffers are used.
This takes about 1.9 seconds to count the primes to a billion (using the fast technique), or about 3.75 clock cycles per culling operation, which is reasonably fast; this is almost 20 times faster the the first naive sieves. As written, the algorithm maintains its efficiency up to about 16 billion and then slows down as the buffer size increases beyond the CPU L1 cache size into the L2 cache size such that it takes about 436.8 seconds to sieve to 100 billion instead of the expected about 300 seconds; however, an extra feature of "double buffered sieving" could be added so that the buffer is sieved in L1 cache slices followed by a final sweep of the entire buffer by the few remaining cull operations that use the larger primes for only a slight reduction in average cycles per cull up to a range of about 2.56e14 (for this CPU). For really large ranges above that, another sieving technique known as the "bucket sieve" that sorts the culling operations by page so that processing time is not expended for values that don't "hit" a given page can be used for only a slight additional reduction in efficiency.
Additionally, maximal wheel factorization can reduce the time by about a factor of four, plus multi-processing where the work is shared across the CPU cores can produce a further speed-up by the factor of the number of cores (only three times on this four-core machine due to the clock speed reducing to 75% of the rate when all cores are used), for an additional about 12 times speed-up for this CPU. These improvements are just slightly too complex to post here.
However, even the version posted shows that the naive "one huge array" implementations should never be used for sieving ranges of over a few million, and that Julia can come very close to the speed of the fastest languages such as C/C++ for the same algorithm.
### Functional Algorithm
One of the best simple purely functional Sieve of Eratosthenes algorithms is the infinite tree folding sequence algorithm as implemented in Haskell. As Julia does not have a standard LazyList implementation or library and as a full memoizing lazy list is not required for this algorithm, the following odds-only code implements the rudiments of a Co-Inductive Stream (CIS) in its implementation:
```julia
const Thunk = Function # can't define other than as a generalized Function
struct CIS{T}
head :: T
tail :: Thunk # produces the next CIS{T}
CIS{T}(head :: T, tail :: Thunk) where T = new(head, tail)
end
Base.eltype(::Type{CIS{T}}) where T = T
Base.IteratorSize(::Type{CIS{T}}) where T = Base.SizeUnknown()
function Base.iterate(C::CIS{T}, state = C) :: Tuple{T, CIS{T}} where T
state.head, state.tail()
end
function treefoldingprimes()::CIS{Int}
function merge(xs::CIS{Int}, ys::CIS{Int})::CIS{Int}
x = xs.head; y = ys.head
if x < y CIS{Int}(x, () -> merge(xs.tail(), ys))
elseif y < x CIS{Int}(y, () -> merge(xs, ys.tail()))
else CIS{Int}(x, () -> merge(xs.tail(), ys.tail())) end
end
function pmultiples(p::Int)::CIS{Int}
adv :: Int = p + p
next(c::Int)::CIS{Int} = CIS{Int}(c, () -> next(c + adv)); next(p * p)
end
function allmultiples(ps::CIS{Int})::CIS{CIS{Int}}
CIS{CIS{Int}}(pmultiples(ps.head), () -> allmultiples(ps.tail()))
end
function pairs(css :: CIS{CIS{Int}})::CIS{CIS{Int}}
nextcss = css.tail()
CIS{CIS{Int}}(merge(css.head, nextcss.head), ()->pairs(nextcss.tail()))
end
function composites(css :: CIS{CIS{Int}})::CIS{Int}
CIS{Int}(css.head.head, ()-> merge(css.head.tail(),
css.tail() |> pairs |> composites))
end
function minusat(n::Int, cs::CIS{Int})::CIS{Int}
if n < cs.head CIS{Int}(n, () -> minusat(n + 2, cs))
else minusat(n + 2, cs.tail()) end
end
oddprimes()::CIS{Int} = CIS{Int}(3, () -> minusat(5, oddprimes()
|> allmultiples |> composites))
CIS{Int}(2, () -> oddprimes())
end
```
when tested with the following:
```julia
@time let count = 0; for p in treefoldingprimes() p > 1000000 && break; count += 1 end; count end
```
it outputs the following:
```txt
1.791058 seconds (10.23 M allocations: 290.862 MiB, 3.64% gc time)
78498
```
At about 1.8 seconds or 4000 cycles per culling operation to calculate the number of primes up to a million, this is very slow, but that is not the fault of Julia but rather just that purely functional incremental Sieve of Eratosthenes implementations are much slower than those using mutable arrays and are only useful over quite limited ranges of a few million. For one thing, incremental algorithms have O(n log n log log n) asymptotic execution complexity rather than O(n log log n) (an extra log n factor) and for another the constant execution overhead is much larger in creating (and garbage collecting) elements in the sequences.
The time for this algorithm is quite comparable to as implemented in other functional languages such as F# and actually faster than implementing the same algorithm in C/C++, but slower than as implemented in purely functional languages such as Haskell or even in only partly functional languages such as Kotlin by a factor of ten or more; this is due to those languages having specialized memory allocation that is very fast at allocating small amounts of memory per allocation as is often a requirement of functional programming. The majority of the time spent for this algorithm is spent allocating memory, and if future versions of Julia are to be of better use in purely functional programming, improvements need to be made to the memory allocation.
===Infinite (Mutable) Iterator Using (Mutable) Dictionary===
To gain some extra speed above the purely functional algorithm above, the Python'ish version as a mutable iterator embedding a mutable standard base Dictionary can be used. The following version uses a secondary delayed injection stream of "base" primes defined recursively to provide the successions of composite values in the Dictionary to be used for sieving:
```Julia
const Prime = UInt64
abstract type PrimesDictAbstract end # used for forward reference
mutable struct PrimesDict <: PrimesDictAbstract
sieve :: Dict{Prime,Prime}
baseprimes :: PrimesDictAbstract
lastbaseprime :: Prime
q :: Prime
PrimesDict() = new(Dict())
end
Base.eltype(::Type{PrimesDict}) = Prime
Base.IteratorSize(::Type{PrimesDict}) = Base.SizeUnknown() # "infinite"...
function Base.iterate(PD::PrimesDict, state::Prime = Prime(0) )
if state < 1
PD.baseprimes = PrimesDict()
PD.lastbaseprime = Prime(3)
PD.q = Prime(9)
return Prime(2), Prime(1)
end
dict = PD.sieve
while true
state += 2
if !haskey(dict, state)
state < PD.q && return state, state
p = PD.lastbaseprime # now, state = PD.q in all cases
adv = p + p # since state is at PD.q, advance to next
dict[state + adv] = adv # adds base prime composite stream
# following initializes secondary base strea first time
p <= 3 && Base.iterate(PD.baseprimes)
p = Base.iterate(PD.baseprimes, p)[1] # next base prime
PD.lastbaseprime = p
PD.q = p * p
else # advance hit composite in dictionary...
adv = pop!(dict, state)
next = state + adv
while haskey(dict, next) next += adv end
dict[next] = adv # past other composite hits in dictionary
end
end
end
```
The above version can be used and tested with similar code as for the functional version, but is about ten times faster at about 400 CPU clock cycles per culling operation, meaning it has a practical range ten times larger although it still has a O(n (log n) (log log n)) asymptotic performance complexity; for larger ranges such as sieving to a billion or more, this is still over a hundred times slower than the page segmented version using a page segmented sieving array.
## Kotlin
```kotlin
fun sieve(limit: Int): List {
val primes = mutableListOf()
if (limit >= 2) {
val numbers = Array(limit + 1) { true }
val sqrtLimit = Math.sqrt(limit.toDouble()).toInt()
for (factor in 2..sqrtLimit) {
if (numbers[factor]) {
for (multiple in (factor * factor)..limit step factor) {
numbers[multiple] = false
}
}
}
numbers.forEachIndexed { number, isPrime ->
if (number >= 2) {
if (isPrime) {
primes.add(number)
}
}
}
}
return primes
}
fun main(args: Array) {
println(sieve(100))
}
```
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]
'''Alternative much faster odds-only version that outputs an enumeration'''
The above version is quite slow for a lot of reasons: It includes even number culling even though those will be eliminated on the first pass; It uses a list rather than an array to do the composite culling (both of the above reasons also meaning it takes more memory); It uses enumerations (for..in) to implement loops at a execution time cost per loop. It also consumes more memory in the final result output as another list.
The following code overcomes most of those problems: It only culls odd composites; it culls a bit-packed primitive array (also saving memory); It uses tailcall recursive functions for the loops, which are compiled into simple loops. It also outputs the results as an enumeration, which isn't fast but does not consume any more memory than the culling array. In this way, the program is only limited in sieving range by the maximum size limit of the culling array, although as it grows larger than the CPU cache sizes, it loses greatly in speed; however, that doesn't matter so much if just enumerating the results.
```kotlin
fun primesOdds(rng: Int): Iterable {
val topi = (rng - 3) shr 1
val lstw = topi shr 5
val sqrtndx = (Math.sqrt(rng.toDouble()).toInt() - 3) shr 1
val cmpsts = IntArray(lstw + 1)
tailrec fun testloop(i: Int) {
if (i <= sqrtndx) {
if (cmpsts[i shr 5] and (1 shl (i and 31)) == 0) {
val p = i + i + 3
tailrec fun cullp(j: Int) {
if (j <= topi) {
cmpsts[j shr 5] = cmpsts[j shr 5] or (1 shl (j and 31))
cullp(j + p)
}
}
cullp((p * p - 3) shr 1)
}
testloop(i + 1)
}
}
tailrec fun test(i: Int): Int {
return if (i <= topi && cmpsts[i shr 5] and (1 shl (i and 31)) != 0) {
test(i + 1)
} else {
i
}
}
testloop(0)
val iter = object : IntIterator() {
var i = -1
override fun nextInt(): Int {
val oi = i
i = test(i + 1)
return if (oi < 0) 2 else oi + oi + 3
}
override fun hasNext() = i < topi
}
return Iterable { -> iter }
}
fun main(args: Array) {
primesOdds(100).forEach { print("$it ") }
println()
println(primesOdds(1000000).count())
}
```
```txt
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
78498
```
'''Concise Functional Versions'''
Ah, one might say, for such a trivial range one writes for conciseness and not for speed. Well, I say, one can still save memory and some time using odds-only and a bit-packed array, but write very clear and concise (but slower) code using nothing but higher order functions and function calling. The following code using such techniques can use the same "main" function for the same output but is about two times slower, mostly due to the extra time spent making (nested) function calls, including the function calls necessary for enumeration. Note that the effect of using the "(l .. h).forEach { .. }" is the same as the "for i in l .. h { .. }" as both use an iteration across the range but the second is just syntax sugar to make it look more imperative:
```kotlin
fun primesOdds(rng: Int): Iterable {
val topi = (rng - 3) / 2 //convert to nearest index
val size = topi / 32 + 1 //word size to include index
val sqrtndx = (Math.sqrt(rng.toDouble()).toInt() - 3) / 2
val cmpsts = IntArray(size)
fun is_p(i: Int) = cmpsts[i shr 5] and (1 shl (i and 0x1F)) == 0
fun cull(i: Int) { cmpsts[i shr 5] = cmpsts[i shr 5] or (1 shl (i and 0x1F)) }
fun cullp(p: Int) = (((p * p - 3) / 2 .. topi).step(p)).forEach { cull(it) }
(0 .. sqrtndx).filter { is_p(it) }.forEach { cullp(it + it + 3) }
fun i2p(i: Int) = if (i < 0) 2 else i + i + 3
val orng = (-1 .. topi).filter { it < 0 || is_p(it) }.map { i2p(it) }
return Iterable { -> orng.iterator() }
}
```
The trouble with the above version is that, at least for Kotlin version 1.0, the ".filter" and ".map" extension functions for Iterable create Java "ArrayList"'s as their output (which are wrapped to return the Kotlin "List" interface), thus take a considerable amount of memory worse than the first version (using an ArrayList to store the resulting primes), since as the calculations are chained to ".map", require a second ArrayList of up to the same size while the mapping is being done. The following version uses Sequences , which aren't backed by any permanent structure, but it is another small factor slower due to the nested function calls:
```kotlin
fun primesOdds(rng: Int): Iterable {
val topi = (rng - 3) / 2 //convert to nearest index
val size = topi / 32 + 1 //word size to include index
val sqrtndx = (Math.sqrt(rng.toDouble()).toInt() - 3) / 2
val cmpsts = IntArray(size)
fun is_p(i: Int) = cmpsts[i shr 5] and (1 shl (i and 0x1F)) == 0
fun cull(i: Int) { cmpsts[i shr 5] = cmpsts[i shr 5] or (1 shl (i and 0x1F)) }
fun iseq(high: Int, low: Int = 0, stp: Int = 1) =
Sequence { (low .. high step(stp)).iterator() }
fun cullp(p: Int) = iseq(topi, (p * p - 3) / 2, p).forEach { cull(it) }
iseq(sqrtndx).filter { is_p(it) }.forEach { cullp(it + it + 3) }
fun i2p(i: Int) = if (i < 0) 2 else i + i + 3
val oseq = iseq(topi, -1).filter { it < 0 || is_p(it) }.map { i2p(it) }
return Iterable { -> oseq.iterator() }
}
```
### Unbounded Versions
'''An incremental odds-only sieve outputting a sequence (iterator)'''
The following Sieve of Eratosthenes is not purely functional in that it uses a Mutable HashMap to store the state of succeeding composite numbers to be skipped over, but embodies the principles of an incremental implementation of the Sieve of Eratosthenes sieving odds-only and is faster than most incremental sieves due to using mutability. As with the fastest of this kind of sieve, it uses a delayed secondary primes feed as a source of base primes to generate the composite number progressions. The code as follows:
```kotlin
fun primesHM(): Sequence = sequence {
yield(2)
fun oddprms(): Sequence = sequence {
yield(3); yield(5) // need at least 2 for initialization
val hm = HashMap()
hm.put(9, 6)
val bps = oddprms().iterator(); bps.next(); bps.next() // skip past 5
yieldAll(generateSequence(SieveState(7, 5, 25)) {
ss ->
var n = ss.n; var q = ss.q
n += 2
while ( n >= q || hm.containsKey(n)) {
if (n >= q) {
val inc = ss.bp shl 1
hm.put(n + inc, inc)
val bp = bps.next(); ss.bp = bp; q = bp * bp
}
else {
val inc = hm.remove(n)!!
var next = n + inc
while (hm.containsKey(next)) {
next += inc
}
hm.put(next, inc)
}
n += 2
}
ss.n = n; ss.q = q
ss
}.map { it.n })
}
yieldAll(oddprms())
}
```
At about 370 clock cycles per culling operation (about 3,800 cycles per prime) on my tablet class Intel CPU, this is not blazing fast but adequate for ranges of a few millions to a hundred million and thus fine for doing things like solving Euler problems. For instance, Euler Problem 10 of summing the primes to two million can be done with the following "one-liner":
```kotlin
primesHM().takeWhile { it <= 2_000_000 }.map { it.toLong() }.sum()
```
to output the correct answer of the following in about 270 milliseconds for my Intel x5-Z8350 at 1.92 Gigahertz:
```txt
142913828922
```
'''A purely functional Incremental Sieve of Eratosthenes that outputs a sequence (iterator)'''
Following is a Kotlin implementation of the Tree Folding Incremental Sieve of Eratosthenes from an adaptation of the algorithm by Richard Bird. It is based on lazy lists, but in fact the memoization (and cost in execution time) of a lazy list is not required and the following code uses a "roll-your-own" implementation of a Co-Inductive Stream CIS). The final output is as a Sequence for convenience in using it. The code is written as purely function in that no mutation is used:
```kotlin>data class CIS CIS) {
fun toSequence() = generateSequence(this) { it.tailf() } .map { it.head }
}
fun primes(): Sequence {
fun merge(a: CIS, b: CIS): CIS {
val ahd = a.head; val bhd = b.head
if (ahd > bhd) return CIS(bhd) { ->merge(a, b.tailf()) }
if (ahd < bhd) return CIS(ahd) { ->merge(a.tailf(), b) }
return CIS(ahd) { ->merge(a.tailf(), b.tailf()) }
}
fun bpmults(p: Int): CIS {
val inc = p + p
fun mlts(c: Int): CIS = CIS(c) { ->mlts(c + inc) }
return mlts(p * p)
}
fun allmults(ps: CIS): CIS> = CIS(bpmults(ps.head)) { allmults(ps.tailf()) }
fun pairs(css: CIS>): CIS> {
val xs = css.head; val yss = css.tailf(); val ys = yss.head
return CIS(merge(xs, ys)) { ->pairs(yss.tailf()) }
}
fun union(css: CIS>): CIS {
val xs = css.head
return CIS(xs.head) { -> merge(xs.tailf(), union(pairs(css.tailf()))) }
}
tailrec fun minus(n: Int, cs: CIS): CIS =
if (n >= cs.head) minus(n + 2, cs.tailf()) else CIS(n) { ->minus(n + 2, cs) }
fun oddprms(): CIS = CIS(3) { -> CIS(5) { ->minus(7, union(allmults(oddprms()))) } }
return CIS(2) { ->oddprms() } .toSequence()
}
fun main(args: Array) {
val limit = 1000000
val strt = System.currentTimeMillis()
println(primes().takeWhile { it <= limit } .count())
val stop = System.currentTimeMillis()
println("Took ${stop - strt} milliseconds.")
}
```
The code is about five times slower than the more imperative hash table based version immediately above due to the costs of the extra levels of function calls in the functional style. The Haskell version from which this is derived is much faster due to the extensive optimizations it does to do with function/closure "lifting" as well as a Garbage Collector specifically tuned for functional code.
'''An unbounded Page Segmented Sieve of Eratosthenes that can output a sequence (iterator)'''
The very fastest implementations of a primes sieve are all based on bit-packed mutable arrays which can be made unbounded by setting them up so that they are a succession of sieved bit-packed arrays that have been culled of composites. The following code is an odds=only implementation that, again, uses a secondary feed of base primes that is only expanded as necessary (in this case memoized by a rudimentary lazy list structure to avoid recalculation for every base primes sweep per page segment):
```kotlin
internal typealias Prime = Long
internal typealias BasePrime = Int
internal typealias BasePrimeArray = IntArray
internal typealias SieveBuffer = ByteArray
// contains a lazy list of a secondary base prime arrays feed
internal data class BasePrimeArrays(val arr: BasePrimeArray,
val rest: Lazy)
: Sequence {
override fun iterator() =
generateSequence(this) { it.rest.value }
.map { it.arr }.iterator()
}
// count the number of zero bits (primes) in a byte array,
fun countComposites(cmpsts: SieveBuffer): Int {
var cnt = 0
for (b in cmpsts) {
cnt += java.lang.Integer.bitCount(b.toInt().and(0xFF))
}
return cmpsts.size.shl(3) - cnt
}
// converts an entire sieved array of bytes into an array of UInt32 primes,
// to be used as a source of base primes...
fun composites2BasePrimeArray(low: Int, cmpsts: SieveBuffer)
: BasePrimeArray {
val lmti = cmpsts.size.shl(3)
val len = countComposites(cmpsts)
val rslt = BasePrimeArray(len)
var j = 0
for (i in 0 until lmti) {
if (cmpsts[i.shr(3)].toInt() and 1.shl(i and 7) == 0) {
rslt[j] = low + i + i; j++
}
}
return rslt
}
// do sieving work based on low starting value for the given buffer and
// the given lazy list of base prime arrays...
fun sieveComposites(low: Prime, buffer: SieveBuffer,
bpas: Sequence) {
val lowi = (low - 3L).shr(1)
val len = buffer.size
val lmti = len.shl(3)
val nxti = lowi + lmti.toLong()
for (bpa in bpas) {
for (bp in bpa) {
val bpi = (bp - 3).shr(1).toLong()
var strti = (bpi * (bpi + 3L)).shl(1) + 3L
if (strti >= nxti) return
val s0 =
if (strti >= lowi) (strti - lowi).toInt()
else {
val r = (lowi - strti) % bp.toLong()
if (r.toInt() == 0) 0 else bp - r.toInt()
}
if (bp <= len.shr(3) && s0 <= lmti - bp.shl(6)) {
val slmti = minOf(lmti, s0 + bp.shl(3))
tailrec fun mods(s: Int) {
if (s < slmti) {
val msk = 1.shl(s and 7)
tailrec fun cull(c: Int) {
if (c < len) {
buffer[c] = (buffer[c].toInt() or msk).toByte()
cull(c + bp)
}
}
cull(s.shr(3)); mods(s + bp)
}
}
mods(s0)
}
else {
tailrec fun cull(c: Int) {
if (c < lmti) {
val w = c.shr(3)
buffer[w] = (buffer[w].toInt() or 1.shl(c and 7)).toByte()
cull(c + bp)
}
}
cull(s0)
}
}
}
}
// starts the secondary base primes feed with minimum size in bits set to 4K...
// thus, for the first buffer primes up to 8293,
// the seeded primes easily cover it as 97 squared is 9409...
fun makeBasePrimeArrays(): Sequence {
var cmpsts = SieveBuffer(512)
fun nextelem(low: Int, bpas: Sequence): BasePrimeArrays {
// calculate size so that the bit span is at least as big as the
// maximum culling prime required, rounded up to minsizebits blocks...
val rqdsz = 2 + Math.sqrt((1 + low).toDouble()).toInt()
val sz = (rqdsz.shr(12) + 1).shl(9) // size iin bytes
if (sz > cmpsts.size) cmpsts = SieveBuffer(sz)
cmpsts.fill(0)
sieveComposites(low.toLong(), cmpsts, bpas)
val arr = composites2BasePrimeArray(low, cmpsts)
val nxt = low + cmpsts.size.shl(4)
return BasePrimeArrays(arr, lazy { ->nextelem(nxt, bpas) })
}
// pre-seeding breaks recursive race,
// as only known base primes used for first page...
var preseedarr = intArrayOf( // pre-seed to 100, can sieve to 10,000...
3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41
, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 )
return BasePrimeArrays(preseedarr, lazy {->nextelem(101, makeBasePrimeArrays())})
}
// a seqence over successive sieved buffer composite arrays,
// returning a tuple of the value represented by the lowest possible prime
// in the sieved composites array and the array itself;
// the array has a 16 Kilobytes minimum size (CPU L1 cache), but
// will grow so that the bit span is larger than the
// maximum culling base prime required, possibly making it larger than
// the L1 cache for large ranges, but still reasonably efficient using
// the L2 cache: very efficient up to about 16e9 range;
// reasonably efficient to about 2.56e14 for two Megabyte L2 cache = > 1 day...
fun makeSievePages(): Sequence> {
val bpas = makeBasePrimeArrays() // secondary source of base prime arrays
fun init(): SieveBuffer {
val c = SieveBuffer(16384); sieveComposites(3L, c, bpas); return c }
return generateSequence(Pair(3L, init())) {
(low, cmpsts) ->
// calculate size so that the bit span is at least as big as the
// max culling prime required, rounded up to minsizebits blocks...
val rqdsz = 2 + Math.sqrt((1 + low).toDouble()).toInt()
val sz = (rqdsz.shr(17) + 1).shl(14) // size iin bytes
val ncmpsts = if (sz > cmpsts.size) SieveBuffer(sz) else cmpsts
ncmpsts.fill(0)
val nlow = low + ncmpsts.size.toLong().shl(4)
sieveComposites(nlow, ncmpsts, bpas)
Pair(nlow, ncmpsts)
}
}
fun countPrimesTo(range: Prime): Prime {
if (range < 3) { if (range < 2) return 0 else return 1 }
var count = 1L
for ((low,cmpsts) in makeSievePages()) {
if (low + cmpsts.size.shl(4) > range) {
val lsti = (range - low).shr(1).toInt()
val lstw = lsti.shr(3)
val msk = -2.shl(lsti.and(7))
count += 32 + lstw.shl(3)
for (i in 0 until lstw)
count -= java.lang.Integer.bitCount(cmpsts[i].toInt().and(0xFF))
count -= java.lang.Integer.bitCount(cmpsts[lstw].toInt().or(msk))
break
} else {
count += countComposites(cmpsts)
}
}
return count
}
// sequence over primes from above page iterator;
// unless doing something special with individual primes, usually unnecessary;
// better to do manipulations based on the composites bit arrays...
// takes at least as long to enumerate the primes as sieve them...
fun primesPaged(): Sequence = sequence {
yield(2L)
for ((low,cmpsts) in makeSievePages()) {
val szbts = cmpsts.size.shl(3)
for (i in 0 until szbts) {
if (cmpsts[i.shr(3)].toInt() and 1.shl(i and 7) != 0) continue
yield(low + i.shl(1).toLong())
}
}
}
```
For this implementation, counting the primes to a million is trivial at about 15 milliseconds on the same CPU as above, or almost too short to count.
It shows its speed in solving the Euler Problem 10 above about five times faster at about 50 milliseconds to give the same output:
It can sum the primes to 200 million or a hundred times the range in just over three seconds.
It finds the count of primes to a billion in about 16 seconds or just about 1000 times slower than to sum the primes to a range 1000 times less for an almost linear response to range as it should be.
However, much of the time (about two thirds) is spent iterating over the results rather than doing the actual work of sieving; for this sort of problem such as counting, finding the nth prime, finding occurrences of maximum prime gaps, etc., one should really use specialized function that directly manipulate the output sieve arrays. Such a function is provided by the `countPrimeTo` function, which can count the primes to a billion (50847534) in about 5.65 seconds, or about 10.6 clock cycles per culling operation or about 210 cycles per prime.
Kotlin isn't really fast even as compared to other virtual machine languages such as C# and F# on CLI but that is mostly due to limitations of the Java Virtual Machine (JVM) as to speed of generated Just In Time (JIT) compilation, handling of primitive number operations, enforced array bounds checks, etc. It will always be much slower than native code producing compilers and the (experimental) native compiler for Kotlin still isn't up to speed (pun intended), producing code that is many times slower than the code run on the JVM (December 2018).
## Liberty BASIC
```lb
'Notice that arrays are globally visible to functions.
'The sieve() function uses the flags() array.
'This is a Sieve benchmark adapted from BYTE 1985
' May, page 286
size = 7000
dim flags(7001)
start = time$("ms")
print sieve(size); " primes found."
print "End of iteration. Elapsed time in milliseconds: "; time$("ms")-start
end
function sieve(size)
for i = 0 to size
if flags(i) = 0 then
prime = i + i + 3
k = i + prime
while k <= size
flags(k) = 1
k = k + prime
wend
sieve = sieve + 1
end if
next i
end function
```
## Limbo
```Go
implement Sieve;
include "sys.m";
sys: Sys;
print: import sys;
include "draw.m";
draw: Draw;
Sieve : module
{
init : fn(ctxt : ref Draw->Context, args : list of string);
};
init (ctxt: ref Draw->Context, args: list of string)
{
sys = load Sys Sys->PATH;
limit := 201;
sieve : array of int;
sieve = array [201] of {* => 1};
(sieve[0], sieve[1]) = (0, 0);
for (n := 2; n < limit; n++) {
if (sieve[n]) {
for (i := n*n; i < limit; i += n) {
sieve[i] = 0;
}
}
}
for (n = 1; n < limit; n++) {
if (sieve[n]) {
print ("%4d", n);
} else {
print(" .");
};
if ((n%20) == 0)
print("\n\n");
}
}
```
## Lingo
```Lingo
-- parent script "sieve"
property _sieve
----------------------------------------
-- @constructor
----------------------------------------
on new (me)
me._sieve = []
return me
end
----------------------------------------
-- Returns list of primes <= n
----------------------------------------
on getPrimes (me, limit)
if me._sieve.count H, K mod H =:= 0 ->
% throw away the multiple we found
L = T1
; % we must not throw away the integer used for filtering
% as we must return a filtered coinductive list
L = [K| T1]
),
filter(H, T, T1).
:- end_object.
```
Example query:
```logtalk
?- sieve::primes(20, P).
P = [2, 3|_S1], % where
_S1 = [5, 7, 11, 13, 17, 19, 2, 3|_S1] .
```
## Lua
```lua
function erato(n)
if n < 2 then return {} end
local t = {0} -- clears '1'
local sqrtlmt = math.sqrt(n)
for i = 2, n do t[i] = 1 end
for i = 2, sqrtlmt do if t[i] ~= 0 then for j = i*i, n, i do t[j] = 0 end end end
local primes = {}
for i = 2, n do if t[i] ~= 0 then table.insert(primes, i) end end
return primes
end
```
The following changes the code to '''odds-only''' using the same large array-based algorithm:
```lua
function erato2(n)
if n < 2 then return {} end
if n < 3 then return {2} end
local t = {}
local lmt = (n - 3) / 2
local sqrtlmt = (math.sqrt(n) - 3) / 2
for i = 0, lmt do t[i] = 1 end
for i = 0, sqrtlmt do if t[i] ~= 0 then
local p = i + i + 3
for j = (p*p - 3) / 2, lmt, p do t[j] = 0 end end end
local primes = {2}
for i = 0, lmt do if t[i] ~= 0 then table.insert(primes, i + i + 3) end end
return primes
end
```
The following code implements '''an odds-only "infinite" generator style using a table as a hash table''', including postponing adding base primes to the table:
```lua
function newEratoInf()
local _cand = 0; local _lstbp = 3; local _lstsqr = 9
local _composites = {}; local _bps = nil
local _self = {}
function _self.next()
if _cand < 9 then if _cand < 1 then _cand = 1; return 2
elseif _cand >= 7 then
--advance aux source base primes to 3...
_bps = newEratoInf()
_bps.next(); _bps.next() end end
_cand = _cand + 2
if _composites[_cand] == nil then -- may be prime
if _cand >= _lstsqr then -- if not the next base prime
local adv = _lstbp + _lstbp -- if next base prime
_composites[_lstbp * _lstbp + adv] = adv -- add cull seq
_lstbp = _bps.next(); _lstsqr = _lstbp * _lstbp -- adv next base prime
return _self.next()
else return _cand end -- is prime
else
local v = _composites[_cand]
_composites[_cand] = nil
local nv = _cand + v
while _composites[nv] ~= nil do nv = nv + v end
_composites[nv] = v
return _self.next() end
end
return _self
end
gen = newEratoInf()
count = 0
while gen.next() <= 10000000 do count = count + 1 end -- sieves to 10 million
print(count)
```
which outputs "664579" in about three seconds. As this code uses much less memory for a given range than the previous ones and retains efficiency better with range, it is likely more appropriate for larger sieve ranges.
## Lucid
prime
where
prime = 2 fby (n whenever isprime(n));
n = 3 fby n+2;
isprime(n) = not(divs) asa divs or prime*prime > N
where
N is current n;
divs = N mod prime eq 0;
end;
end
### recursive
sieve( N )
where
N = 2 fby N + 1;
sieve( i ) =
i fby sieve ( i whenever i mod first i ne 0 ) ;
end
## M2000 Interpreter
```M2000 Interpreter
Module EratosthenesSieve (x) {
\\ Κόσκινο του Ερατοσθένη
If x>200000 Then Exit
Dim i(x+1)
k=2
k2=x div 2
While k<=k2 {
m=k+k
While m<=x {
i(m)=1
m+=k
}
k++
}
For i=2 to x {
If i(i)=0 Then Print i,
}
Print
}
EratosthenesSieve 1000
```
## M4
```M4
define(`lim',100)dnl
define(`for',
`ifelse($#,0,
``$0'',
`ifelse(eval($2<=$3),1,
`pushdef(`$1',$2)$5`'popdef(`$1')$0(`$1',eval($2+$4),$3,$4,`$5')')')')dnl
for(`j',2,lim,1,
`ifdef(a[j],
`',
`j for(`k',eval(j*j),lim,j,
`define(a[k],1)')')')
```
Output:
```txt
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
```
## Maple
```Maple
Eratosthenes := proc(n::posint)
local numbers_to_check, i, k;
numbers_to_check := [seq(2 .. n)];
for i from 2 to floor(sqrt(n)) do
for k from i by i while k <= n do
if evalb(k <> i) then
numbers_to_check[k - 1] := 0;
end if;
end do;
end do;
numbers_to_check := remove(x -> evalb(x = 0), numbers_to_check);
return numbers_to_check;
end proc:
```
```txt
Eratosthenes(100);
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]
```
## Mathematica
```Mathematica
Eratosthenes[n_] := Module[{numbers = Range[n]},
Do[If[numbers[[i]] != 0, Do[numbers[[i j]] = 0, {j, 2, n/i}]], {i,
2, Sqrt[n]}];
Select[numbers, # > 1 &]]
Eratosthenes[100]
```
### Slightly Optimized Version
The below has been improved to not require so many operations per composite number cull for about two thirds the execution time:
```Mathematica
Eratosthenes[n_] := Module[{numbers = Range[n]},
Do[If[numbers[[i]] != 0, Do[numbers[[j]] = 0, {j,i i,n,i}]],{i,2,Sqrt[n]}];
Select[numbers, # > 1 &]]
Eratosthenes[100]
```
===Sieving Odds-Only Version===
The below has been further improved to only sieve odd numbers for a further reduction in execution time by a factor of over two:
```Mathematica
Eratosthenes2[n_] := Module[{numbers = Range[3, n, 2], limit = (n - 1)/2},
Do[c = numbers[[i]]; If[c != 0,
Do[numbers[[j]] = 0, {j,(c c - 1)/2,limit,c}]], {i,1,(Sqrt[n] - 1)/2}];
Prepend[Select[numbers, # > 1 &], 2]]
Eratosthenes2[100]
```
## MATLAB
### Somewhat optimized true Sieve of Eratosthenes
```MATLAB
function P = erato(x) % Sieve of Eratosthenes: returns all primes between 2 and x
P = [0 2:x]; % Create vector with all ints between 2 and x where
% position 1 is hard-coded as 0 since 1 is not a prime.
for n = 2:sqrt(x) % All primes factors lie between 2 and sqrt(x).
if P(n) % If the current value is not 0 (i.e. a prime),
P(n*n:n:x) = 0; % then replace all further multiples of it with 0.
end
end % At this point P is a vector with only primes and zeroes.
P = P(P ~= 0); % Remove all zeroes from P, leaving only the primes.
end
```
The optimization lies in fewer steps in the for loop, use of MATLAB's built-in array operations and no modulo calculation.
'''Limitation:''' your machine has to be able to allocate enough memory for an array of length x.
### A more efficient Sieve
A more efficient Sieve avoids creating a large double precision vector P, instead using a logical array (which consumes 1/8 the memory of a double array of the same size) and only converting to double those values corresponding to primes.
```MATLAB
function P = sieveOfEratosthenes(x)
ISP = [false true(1, x-1)]; % 1 is not prime, but we start off assuming all numbers between 2 and x are
for n = 2:sqrt(x)
if ISP(n)
ISP(n*n:n:x) = false; % Multiples of n that are greater than n*n are not primes
end
end
% The ISP vector that we have calculated is essentially the output of the ISPRIME function called on 1:x
P = find(ISP); % Convert the ISPRIME output to the values of the primes by finding the locations
% of the TRUE values in S.
end
```
You can compare the output of this function against the PRIMES function included in MATLAB, which performs a somewhat more memory-efficient Sieve (by not storing even numbers, at the expense of a more complicated indexing expression inside the IF statement.)
## Maxima
```maxima
sieve(n):=block(
[a:makelist(true,n),i:1,j],
a[1]:false,
do (
i:i+1,
unless a[i] do i:i+1,
if i*i>n then return(sublist_indices(a,identity)),
for j from i*i step i while j<=n do a[j]:false
)
)$
```
## MAXScript
fn eratosthenes n =
(
multiples = #()
print 2
for i in 3 to n do
(
if (findItem multiples i) == 0 then
(
print i
for j in (i * i) to n by i do
(
append multiples j
)
)
)
)
eratosthenes 100
## Mercury
```Mercury
:- module sieve.
:- interface.
:- import_module io.
:- pred main(io::di, io::uo) is det.
:- implementation.
:- import_module bool, array, int.
main(!IO) :-
sieve(50, Sieve),
dump_primes(2, size(Sieve), Sieve, !IO).
:- pred dump_primes(int, int, array(bool), io, io).
:- mode dump_primes(in, in, array_di, di, uo) is det.
dump_primes(N, Limit, !.A, !IO) :-
( if N < Limit then
unsafe_lookup(!.A, N, Prime),
(
Prime = yes,
io.write_line(N, !IO)
;
Prime = no
),
dump_primes(N + 1, Limit, !.A, !IO)
else
true
).
:- pred sieve(int, array(bool)).
:- mode sieve(in, array_uo) is det.
sieve(N, !:A) :-
array.init(N, yes, !:A),
sieve(2, N, !A).
:- pred sieve(int, int, array(bool), array(bool)).
:- mode sieve(in, in, array_di, array_uo) is det.
sieve(N, Limit, !A) :-
( if N < Limit then
unsafe_lookup(!.A, N, Prime),
(
Prime = yes,
sift(N + N, N, Limit, !A),
sieve(N + 1, Limit, !A)
;
Prime = no,
sieve(N + 1, Limit, !A)
)
else
true
).
:- pred sift(int, int, int, array(bool), array(bool)).
:- mode sift(in, in, in, array_di, array_uo) is det.
sift(I, N, Limit, !A) :-
( if I < Limit then
unsafe_set(I, no, !A),
sift(I + N, N, Limit, !A)
else
true
).
```
## Microsoft Small Basic
```microsoftsmallbasic
TextWindow.Write("Enter number to search to: ")
limit = TextWindow.ReadNumber()
For n = 2 To limit
flags[n] = 0
EndFor
For n = 2 To math.SquareRoot(limit)
If flags[n] = 0 Then
For K = n * n To limit Step n
flags[K] = 1
EndFor
EndIf
EndFor
' Display the primes
If limit >= 2 Then
TextWindow.Write(2)
For n = 3 To limit
If flags[n] = 0 Then
TextWindow.Write(", " + n)
EndIf
EndFor
TextWindow.WriteLine("")
EndIf
```
=={{header|Modula-3}}==
### Regular version
This version runs slow because of the way I/O is implemented in the CM3 compiler. Setting ListPrimes = FALSE achieves speed comparable to C on sufficiently high values of LastNum (e.g., 10^6).
```modula3
MODULE Eratosthenes EXPORTS Main;
IMPORT IO;
FROM Math IMPORT sqrt;
CONST
LastNum = 1000;
ListPrimes = TRUE;
VAR
a: ARRAY[2..LastNum] OF BOOLEAN;
VAR
n := LastNum - 2 + 1;
BEGIN
(* set up *)
FOR i := FIRST(a) TO LAST(a) DO
a[i] := TRUE;
END;
(* declare a variable local to a block *)
VAR b := FLOOR(sqrt(FLOAT(LastNum, LONGREAL)));
(* the block must follow immediately *)
BEGIN
(* print primes and mark out composites up to sqrt(LastNum) *)
FOR i := FIRST(a) TO b DO
IF a[i] THEN
IF ListPrimes THEN IO.PutInt(i); IO.Put(" "); END;
FOR j := i*i TO LAST(a) BY i DO
IF a[j] THEN
a[j] := FALSE;
DEC(n);
END;
END;
END;
END;
(* print remaining primes *)
IF ListPrimes THEN
FOR i := b + 1 TO LAST(a) DO
IF a[i] THEN
IO.PutInt(i); IO.Put(" ");
END;
END;
END;
END;
(* report *)
IO.Put("There are "); IO.PutInt(n);
IO.Put(" primes from 2 to "); IO.PutInt(LastNum);
IO.PutChar('\n');
END Eratosthenes.
```
### Advanced version
This version uses more "advanced" types.
```modula3
(* From the CM3 examples folder (comments removed). *)
MODULE Sieve EXPORTS Main;
IMPORT IO;
TYPE
Number = [2..1000];
Set = SET OF Number;
VAR
prime: Set := Set {FIRST(Number) .. LAST(Number)};
BEGIN
FOR i := FIRST(Number) TO LAST(Number) DO
IF i IN prime THEN
IO.PutInt(i);
IO.Put(" ");
FOR j := i TO LAST(Number) BY i DO
prime := prime - Set{j};
END;
END;
END;
IO.Put("\n");
END Sieve.
```
## MUMPS
```MUMPS
ERATO1(HI)
;performs the Sieve of Erotosethenes up to the number passed in.
;This version sets an array containing the primes
SET HI=HI\1
KILL ERATO1 ;Don't make it new - we want it to remain after we quit the function
NEW I,J,P
FOR I=2:1:(HI**.5)\1 FOR J=I*I:I:HI SET P(J)=1
FOR I=2:1:HI S:'$DATA(P(I)) ERATO1(I)=I
KILL I,J,P
QUIT
```
Example:
```txt
USER>SET MAX=100,C=0 DO ERATO1^ROSETTA(MAX)
USER>WRITE !,"PRIMES BETWEEN 1 AND ",MAX,! FOR SET I=$ORDER(ERATO1(I)) Q:+I<1 WRITE I,", "
PRIMES BETWEEN 1 AND 100
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73,79, 83, 89, 97,
```
## NetRexx
===Version 1 (slow)===
```Rexx
/* NetRexx */
options replace format comments java crossref savelog symbols binary
parse arg loWatermark hiWatermark .
if loWatermark = '' | loWatermark = '.' then loWatermark = 1
if hiWatermark = '' | hiWatermark = '.' then hiWatermark = 200
do
if \loWatermark.datatype('w') | \hiWatermark.datatype('w') then -
signal NumberFormatException('arguments must be whole numbers')
if loWatermark > hiWatermark then -
signal IllegalArgumentException('the start value must be less than the end value')
seive = sieveOfEratosthenes(hiWatermark)
primes = getPrimes(seive, loWatermark, hiWatermark).strip
say 'List of prime numbers from' loWatermark 'to' hiWatermark 'via a "Sieve of Eratosthenes" algorithm:'
say ' 'primes.changestr(' ', ',')
say ' Count of primes:' primes.words
catch ex = Exception
ex.printStackTrace
end
return
method sieveOfEratosthenes(hn = long) public static binary returns Rexx
sv = Rexx(isTrue)
sv[1] = isFalse
ix = long
jx = long
loop ix = 2 while ix * ix <= hn
if sv[ix] then loop jx = ix * ix by ix while jx <= hn
sv[jx] = isFalse
end jx
end ix
return sv
method getPrimes(seive = Rexx, lo = long, hi = long) private constant binary returns Rexx
primes = Rexx('')
loop p_ = lo to hi
if \seive[p_] then iterate p_
primes = primes p_
end p_
return primes
method isTrue public constant binary returns boolean
return 1 == 1
method isFalse public constant binary returns boolean
return \isTrue
```
;Output
List of prime numbers from 1 to 200 via a "Sieve of Eratosthenes" algorithm:
2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,191,193,197,199
Count of primes: 46
```
===Version 2 (significantly, i.e. 10 times faster)===
```NetRexx
/* NetRexx ************************************************************
* Essential improvements:Use boolean instead of Rexx for sv
* and remove methods isTrue and isFalse
* 24.07.2012 Walter Pachl courtesy Kermit Kiser
**********************************************************************/
options replace format comments java crossref savelog symbols binary
parse arg loWatermark hiWatermark .
if loWatermark = '' | loWatermark = '.' then loWatermark = 1
if hiWatermark = '' | hiWatermark = '.' then hiWatermark = 200000
startdate=Date Date()
do
if \loWatermark.datatype('w') | \hiWatermark.datatype('w') then -
signal NumberFormatException('arguments must be whole numbers')
if loWatermark > hiWatermark then -
signal IllegalArgumentException(-
'the start value must be less than the end value')
sieve = sieveOfEratosthenes(hiWatermark)
primes = getPrimes(sieve, loWatermark, hiWatermark).strip
if hiWatermark = 200 Then do
say 'List of prime numbers from' loWatermark 'to' hiWatermark
say ' 'primes.changestr(' ', ',')
end
catch ex = Exception
ex.printStackTrace
end
enddate=Date Date()
Numeric Digits 20
say (enddate.getTime-startdate.getTime)/1000 'seconds elapsed'
say ' Count of primes:' primes.words
return
method sieveOfEratosthenes(hn = int) -
public static binary returns boolean[]
true = boolean 1
false = boolean 0
sv = boolean[hn+1]
sv[1] = false
ix = int
jx = int
loop ix=2 to hn
sv[ix]=true
end ix
loop ix = 2 while ix * ix <= hn
if sv[ix] then loop jx = ix * ix by ix while jx <= hn
sv[jx] = false
end jx
end ix
return sv
method getPrimes(sieve = boolean[], lo = int, hi = int) -
private constant binary Returns Rexx
p_ = int
primes = Rexx('')
loop p_ = lo to hi
if \sieve[p_] then iterate p_
primes = primes p_
end p_
return primes
```
## Nial
primes is sublist [ each (2 = sum eachright (0 = mod) [pass,count]), pass ] rest count
Using it
|primes 10
=2 3 5 7
## Nim
```nim
from math import sqrt
iterator primesUpto(limit: int): int =
let sqrtLimit = int(sqrt(float64(limit)))
var composites = newSeq[bool](limit + 1)
for n in 2 .. sqrtLimit: # cull to square root of limit
if not composites[n]: # if prime -> cull its composites
for c in countup(n *% n, limit +% 1, n): # start at ``n`` squared
composites[c] = true
for n in 2 .. limit: # separate iteration over results
if not composites[n]:
yield n
stdout.write "The primes up to 100 are: "
for x in primesUpto(100):
stdout.write(x, " ")
echo ""
var count = 0
for p in primesUpto(1000000):
count += 1
echo "There are ", count, " primes up to 1000000."
```
```txt
Primes are:
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
There are 78498 primes up to 1000000.
```
'''Alternate odds-only bit-packed version'''
The above version wastes quite a lot of memory by using a sequence of boolean values to sieve the composite numbers and sieving all numbers when two is the only even prime. The below code uses a bit-packed sequence to save a factor of eight in memory and also sieves only odd primes for another memory saving by a factor of two; it is also over two and a half times faster due to reduced number of culling operations and better use of the CPU cache as a little cache goes a lot further - this better use of cache is more than enough to make up for the extra bit-packing shifting operations:
```nim
iterator isoe_upto(top: uint): uint =
let topndx = int((top - 3) div 2)
let sqrtndx = (int(sqrt float64(top)) - 3) div 2
var cmpsts = newSeq[uint32](topndx div 32 + 1)
for i in 0 .. sqrtndx: # cull composites for primes
if (cmpsts[i shr 5] and (1u32 shl (i and 31))) == 0:
let p = i + i + 3
for j in countup((p * p - 3) div 2, topndx, p):
cmpsts[j shr 5] = cmpsts[j shr 5] or (1u32 shl (j and 31))
yield 2 # separate culling above and iteration here
for i in 0 .. topndx:
if (cmpsts[i shr 5] and (1u32 shl (i and 31))) == 0:
yield uint(i + i + 3)
```
The above code can be used with the same output functions as in the first code, just replacing the name of the iterator "iprimes_upto" with this iterator's name "isoe_upto" in two places. The output will be identical.
=
## Nim Unbounded Versions
=
For many purposes, one doesn't know the exact upper limit desired to easily use the above versions; in addition, those versions use an amount of memory proportional to the range sieved. In contrast, unbounded versions continuously update their range as they progress and only use memory proportional to the secondary base primes stream, which is only proportional to the square root of the range. One of the most basic functional versions is the TreeFolding sieve which is based on merging lazy streams as per Richard Bird's contribution to incremental sieves in Haskell, but which has a much better asymptotic execution complexity due to the added tree folding. The following code is a version of that in Nim (odds-only):
```nim
from times import epochTime
type PrimeType = int
iterator primesTreeFolding(): PrimeType {.closure.} =
# needs a Co Inductive Stream - CIS...
type
CIS[T] = ref object
head: T
tail: () -> CIS[T]
proc merge(xs, ys: CIS[PrimeType]): CIS[PrimeType] =
let x = xs.head; let y = ys.head
if x < y:
CIS[PrimeType](head: x, tail: () => merge(xs.tail(), ys))
elif y < x:
CIS[PrimeType](
head: y,
tail: () => merge(xs, ys.tail()))
else:
CIS[PrimeType](
head: x,
tail: () => merge(xs.tail(), ys.tail()))
proc pmults(p: PrimeType): CIS[PrimeType] =
let inc = p + p
proc mlts(c: PrimeType): CIS[PrimeType] =
CIS[PrimeType](head: c, tail: () => mlts(c + inc))
mlts(p * p)
proc allmults(ps: CIS[PrimeType]): CIS[CIS[PrimeType]] =
CIS[CIS[PrimeType]](
head: pmults(ps.head),
tail: () => allmults(ps.tail()))
proc pairs(css: CIS[CIS[PrimeType]]): CIS[CIS[PrimeType]] =
let cs0 = css.head; let rest0 = css.tail()
CIS[CIS[PrimeType]](
head: merge(cs0, rest0.head),
tail: () => pairs(rest0.tail()))
proc cmpsts(css: CIS[CIS[PrimeType]]): CIS[PrimeType] =
let cs0 = css.head
CIS[PrimeType](
head: cs0.head,
tail: () => merge(cs0.tail(), css.tail().pairs.cmpsts))
proc minusAt(n: PrimeType, cs: CIS[PrimeType]): CIS[PrimeType] =
var nn = n; var ncs = cs
while nn >= ncs.head: nn += 2; ncs = ncs.tail()
CIS[PrimeType](head: nn, tail: () => minusAt(nn + 2, ncs))
proc oddprms(): CIS[PrimeType] =
CIS[PrimeType](
head: 3.PrimeType,
tail: () => minusAt(5.PrimeType, oddprms().allmults.cmpsts))
var prms =
CIS[PrimeType](head: 2.PrimeType, tail: () => oddprms())
while true: yield prms.head; prms = prms.tail()
stdout.write "The first 25 primes are: "
var counter = 0
for p in primesTreeFolding():
if counter >= 25: break
stdout.write(p, " "); counter += 1
echo ""
start = epochTime()
counter = 0
for p in primesTreeFolding():
if p > range: break else: counter += 1
elapsed = epochTime() - start
echo "There are ", counter, " primes up to 1000000."
echo "This test took ", elapsed, " seconds."
```
```txt
The first 25 primes are: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
There are 78498 primes up to 1000000
This test took 1.753238677978516 seconds.
```
It takes in the order of a second to compute the primes to a million (depending on CPU), and is so slow due to the many small memory allocations/de-allocations required, which is a characteristic of functional forms of code. Is is purely functional in that everything is immutable other than that Nim does not have Tail Call Optimization (TCO) so that we can freely use function recursion with no execution time cost; therefore, where necessary this is implemented with imperative loops, which is what TCO is generally turned into such forms "under the covers". It is also slow due to the algorithm being only O(n (log n) (log (log n))) rather than without the extra "log n" factor as some version have. This slowness makes it only moderately useful for ranges up to a few million.
Since the algorithm does not require the memoization of a full lazy list, it uses an internal Co Inductive Stream of deferred execution states, finally outputting an iterator to enumerate over the lazily computed stream of primes.
'''A faster alternative using a mutable hash table (odds-only)'''
To show the cost of functional forms of code, the following code is written embracing mutability, both by using a mutable hash table to store the state of incremental culling by the secondary stream of base primes and by using mutable values to store the state wherever possible, as per the following code:
```nim
import tables
type PrimeType = int
proc primesHashTable(): iterator(): PrimeType {.closure.} =
iterator output(): PrimeType {.closure.} =
# some initial values to avoid race and reduce initializations...
yield 2.PrimeType; yield 3.PrimeType; yield 5.PrimeType; yield 7.PrimeType
var h = initTable[PrimeType,PrimeType]()
var n = 9.PrimeType
let bps = primesHashTable()
var bp = bps() # advance past 2
bp = bps(); var q = bp * bp # to initialize with 3
while true:
if n >= q:
let inc = bp + bp
h.add(n + inc, inc)
bp = bps(); q = bp * bp
elif h.hasKey(n):
var inc: PrimeType
discard h.take(n, inc)
var nxt = n + inc
while h.hasKey(nxt): nxt += inc # ensure no duplicates
h.add(nxt, inc)
else: yield n
n += 2.PrimeType
output
stdout.write "The first 25 primes are: "
var counter = 0
var iter = primesHashTable()
for p in iter():
if counter >= 25: break else: stdout.write(p, " "); counter += 1
echo ""
let start = epochTime()
counter = 0
iter = primesHashTable()
for p in iter():
if p > 1000000: break else: counter += 1
let elapsed = epochTime() - start
echo "The number of primes up to a million is: ", counter
stdout.write("This test took ", elapsed, " seconds.\n")
```
The output is identical to the first unbounded version, other than it is over about 15 times faster sieving to a million. For larger ranges it will continue to pull further ahead of the above version due to only O(n (log (log n))) performance because of the hash table having an average of O(1) access, and it is only so slow due to the large constant overhead of doing the hashing calculations and look-ups.
'''Very fast Page Segmented version using a bit-packed mutable array (odds-only)'''
Note: This version is used as a very fast alternative in [[Extensible_prime_generator#Nim]]
For the highest speeds, one needs to use page segmented mutable arrays as in the bit-packed version here:
```nim
# a Page Segmented Odd-Only Bit-Packed Sieve of Eratosthenes...
from times import epochTime # for testing
from bitops import popCount
type Prime = uint64
let LIMIT = 1_000_000_000.Prime
let CPUL1CACHE = 16384 # in bytes
const FRSTSVPRM = 3.Prime
type
BasePrime = uint32
BasePrimeArray = seq[BasePrime]
SieveBuffer = seq[byte] # byte size gives the most potential efficiency...
# define a general purpose lazy list to use as secondary base prime arrays feed
# NOT thread safe; needs a Mutex gate to make it so, but not threaded (yet)...
type
BasePrimeArrayLazyList = ref object
head: BasePrimeArray
tailf: proc (): BasePrimeArrayLazyList {.closure.}
tail: BasePrimeArrayLazyList
template makeBasePrimeArrayLazyList(hd: BasePrimeArray;
body: untyped): untyped = # factory constructor
let thnk = proc (): BasePrimeArrayLazyList {.closure.} = body
BasePrimeArrayLazyList(head: hd, tailf: thnk)
proc rest(lzylst: sink BasePrimeArrayLazyList): BasePrimeArrayLazyList {.inline.} =
if lzylst.tailf != nil: lzylst.tail = lzylst.tailf(); lzylst.tailf = nil
return lzylst.tail
iterator items(lzylst: BasePrimeArrayLazyList): BasePrime {.inline.} =
var ll = lzylst
while ll != nil:
for bp in ll.head: yield bp
ll = ll.rest
# count the number of zero bits (primes) in a SieveBuffer,
# uses native popCount for extreme speed;
# counts up to the bit index of the last bit to be counted...
proc countSieveBuffer(lsti: int; cmpsts: SieveBuffer): int =
let lstw = (lsti shr 3) and -8; let lstm = lsti and 63 # last word and bit index!
result = (lstw shl 3) + 64 # preset for all ones!
let cmpstsa = cast[int](cmpsts[0].unsafeAddr)
let cmpstslsta = cmpstsa + lstw
for csa in countup(cmpstsa, cmpstslsta - 1, 8):
result -= cast[ptr uint64](csa)[].popCount # subtract number of found ones!
let msk = (0'u64 - 2'u64) shl lstm # mask for the unused bits in last word!
result -= (cast[ptr uint64](cmpstslsta)[] or msk).popCount
# a fast fill SieveBuffer routine using pointers...
proc fillSieveBuffer(sb: var SieveBuffer) = zeroMem(sb[0].unsafeAddr, sb.len)
const BITMASK = [1'u8, 2, 4, 8, 16, 32, 64, 128] # faster than shifting!
# do sieving work, based on low starting value for the given buffer and
# the given lazy list of base prime arrays...
proc cullSieveBuffer(lwi: int; bpas: BasePrimeArrayLazyList;
sb: var SieveBuffer) =
let len = sb.len; let szbits = len shl 3; let nxti = lwi + szbits
for bp in bpas:
let bpwi = ((bp.Prime - FRSTSVPRM) shr 1).int
var s = (bpwi shl 1) * (bpwi + FRSTSVPRM.int) + FRSTSVPRM.int
if s >= nxti: break
if s >= lwi: s -= lwi
else:
let r = (lwi - s) mod bp.int
s = (if r == 0: 0 else: bp.int - r)
let clmt = szbits - (bp.int shl 3)
# if len == CPUL1CACHE: continue
if s < clmt:
let slmt = s + (bp.int shl 3)
while s < slmt:
let msk = BITMASK[s and 7]
for c in countup(s shr 3, len - 1, bp.int):
sb[c] = sb[c] or msk
s += bp.int
continue
while s < szbits:
let w = s shr 3; sb[w] = sb[w] or BITMASK[s and 7]; s += bp.int # (1'u8 shl (s and 7))
proc makeBasePrimeArrays(): BasePrimeArrayLazyList # forward reference!
# an iterator over successive sieved buffer composite arrays,
# returning whatever type the cnvrtr produces from
# the low index and the culled SieveBuffer...
proc makePrimePages[T](
strtwi, sz: int; cnvrtrf: proc (li: int; sb: var SieveBuffer): T {.closure.}
): (iterator(): T {.closure.}) =
var lwi = strtwi; let bpas = makeBasePrimeArrays(); var cmpsts = newSeq[byte](sz)
return iterator(): T {.closure.} =
while true:
fillSieveBuffer(cmpsts); cullSieveBuffer(lwi, bpas, cmpsts)
yield cnvrtrf(lwi, cmpsts); lwi += cmpsts.len shl 3
# starts the secondary base primes feed with minimum size in bits set to 4K...
# thus, for the first buffer primes up to 8293,
# the seeded primes easily cover it as 97 squared is 9409.
proc makeBasePrimeArrays(): BasePrimeArrayLazyList =
# converts an entire sieved array of bytes into an array of base primes,
# to be used as a source of base primes as part of the Lazy List...
proc sb2bpa(li: int; sb: var SieveBuffer): BasePrimeArray =
let szbits = sb.len shl 3; let len = countSieveBuffer(szbits - 1, sb)
result = newSeq[BasePrime](len); var j = 0
for i in 0 ..< szbits:
if (sb[i shr 3] and BITMASK[i and 7]) == 0'u8:
result[j] = FRSTSVPRM.BasePrime + ((li + i) shl 1).BasePrime; j.inc
proc nxtbparr(
pgen: iterator (): BasePrimeArray {.closure.}): BasePrimeArrayLazyList =
return makeBasePrimeArrayLazyList(pgen()): nxtbparr(pgen)
# pre-seeding first array breaks recursive race,
# dummy primes of all odd numbers starting at FRSTSVPRM (unculled)...
var cmpsts = newSeq[byte](512)
let dummybparr = sb2bpa(0, cmpsts)
let fakebps = makeBasePrimeArrayLazyList(dummybparr): nil # used just once here!
cullSieveBuffer(0, fakebps, cmpsts)
return makeBasePrimeArrayLazyList(sb2bpa(0, cmpsts)):
nxtbparr(makePrimePages(4096, 512, sb2bpa)) # lazy recursive call breaks race!
# iterator over primes from above page iterator;
# takes at least as long to enumerate the primes as sieve them...
iterator primesPaged(): Prime {.inline.} =
yield 2
proc mkprmarr(li: int; sb: var SieveBuffer): seq[Prime] =
let szbits = sb.len shl 3; let low = FRSTSVPRM + (li + li).Prime; var j = 0
let len = countSieveBuffer(szbits - 1, sb); result = newSeq[Prime](len)
for i in 0 ..< szbits:
if (sb[i shr 3] and BITMASK[i and 7]) == 0'u8:
result[j] = low + (i + i).Prime; j.inc
let gen = makePrimePages(0, CPUL1CACHE, mkprmarr)
for prmpg in gen():
for prm in prmpg: yield prm
proc countPrimesTo(range: Prime): int64 =
if range < FRSTSVPRM: return (if range < 2: 0 else: 1)
result = 1; let rngi = ((range - FRSTSVPRM) shr 1).int
proc cntr(li: int; sb: var SieveBuffer): (int, int) {.closure.} =
let szbits = sb.len shl 3; let nxti = li + szbits; result = (0, nxti)
if nxti <= rngi: result[0] += countSieveBuffer(szbits - 1, sb)
else: result[0] += countSieveBuffer(rngi - li, sb)
let gen = makePrimePages(0, CPUL1CACHE, cntr)
for count, nxti in gen():
result += count; if nxti > rngi: break
# showing results...
echo "Page Segmented Bit-Packed Odds-Only Sieve of Eratosthenes"
echo "Needs at least ", CPUL1CACHE, " bytes of CPU L1 cache memory.\n"
stdout.write "First 25 primes: "
var counter0 = 0
for p in primesPaged():
if counter0 >= 25: break
stdout.write(p, " "); counter0.inc
echo ""
stdout.write "The number of primes up to a million is: "
var counter1 = 0
for p in primesPaged():
if p > 1_000_000.Prime: break else: counter1.inc
stdout.write counter1, " - these both found by (slower) enumeration.\n"
let start = epochTime()
#[ # slow way to count primes takes as long to enumerate as sieve!
var counter = 0
for p in primesPaged():
if p > LIMIT: break else: counter.inc
# ]#
let counter = countPrimesTo LIMIT # the fast way using native popCount!
let elpsd = epochTime() - start
echo "Found ", counter, " primes up to ", LIMIT, " in ", elpsd, " seconds."
```
```txt
Page Segmented Bit-Packed Odds-Only Sieve of Eratosthenes
Needs at least 16384 bytes of CPU L1 cache memory.
First 25 primes: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
The number of primes up to a million is: 78498 - these both found by (slower) enumeration.
Found 50847534 primes up to 1000000000 in 1.399795055389404 seconds.
```
The above version approaches a hundred times faster than the incremental style versions above due to the high efficiency of direct mutable memory operations in modern CPU's, and is useful for ranges of billions. This version maintains its efficiency using the CPU L1 cache to a range of over 16 billion and then gets a little slower for ranges of a trillion or more using the CPU's L2 cache. It takes an average of only about 3.5 CPU clock cycles per composite number cull, or about 70 CPU clock cycles per prime found.
Note that the fastest performance is realized by using functions that directly manipulate the output "seq" (array) of culled bit number representations such as the `countPrimesTo` function provided, as enumeration using the `primesPaged` iterator takes about as long to enumerate the found primes as it takes to cull the composites.
Many further improvements in speed can be made, as in tuning the medium ranges to more efficiently use the CPU caches for an improvement in the middle ranges of up to a factor of about two, full maximum wheel factorization for a further improvement of about four, extreme loop unrolling for a further improvement of approximately two, multi-threading for an improvement of the factor of effective CPU cores used, etc. However, these improvements are of little point when used with enumeration; for instance, if one successfully reduced the time to sieve the composite numbers to zero, it would still take about a second just to enumerate the resulting primes over a range of a billion.
## Niue
```Niue
[ dup 2 < ] '<2 ;
[ 1 + 'count ; [ <2 [ , ] when ] count times ] 'fill-stack ;
0 'n ; 0 'v ;
[ .clr 0 'n ; 0 'v ; ] 'reset ;
[ len 1 - n - at 'v ; ] 'set-base ;
[ n 1 + 'n ; ] 'incr-n ;
[ mod 0 = ] 'is-factor ;
[ dup * ] 'sqr ;
[ set-base
v sqr 2 at > not
[ [ dup v = not swap v is-factor and ] remove-if incr-n run ] when ] 'run ;
[ fill-stack run ] 'sieve ;
( tests )
10 sieve .s ( => 2 3 5 7 9 ) reset newline
30 sieve .s ( => 2 3 5 7 11 13 17 19 23 29 )
```
=={{header|Oberon-2}}==
```oberon2
MODULE Primes;
IMPORT Out, Math;
CONST N = 1000;
VAR a: ARRAY N OF BOOLEAN;
i, j, m: INTEGER;
BEGIN
(* Set all elements of a to TRUE. *)
FOR i := 1 TO N - 1 DO
a[i] := TRUE;
END;
(* Compute square root of N and convert back to INTEGER. *)
m := ENTIER(Math.Sqrt(N));
FOR i := 2 TO m DO
IF a[i] THEN
FOR j := 2 TO (N - 1) DIV i DO
a[i*j] := FALSE;
END;
END;
END;
(* Print all the elements of a that are TRUE. *)
FOR i := 2 TO N - 1 DO
IF a[i] THEN
Out.Int(i, 4);
END;
END;
Out.Ln;
END Primes.
```
## OCaml
### Imperative
```ocaml
let sieve n =
let is_prime = Array.create n true in
let limit = truncate(sqrt (float (n - 1))) in
for i = 2 to limit do
if is_prime.(i) then
let j = ref (i*i) in
while !j < n do
is_prime.(!j) <- false;
j := !j + i;
done
done;
is_prime.(0) <- false;
is_prime.(1) <- false;
is_prime
```
```ocaml
let primes n =
let primes, _ =
let sieve = sieve n in
Array.fold_right
(fun is_prime (xs, i) -> if is_prime then (i::xs, i-1) else (xs, i-1))
sieve
([], Array.length sieve - 1)
in
primes
```
in the top-level:
# primes 100 ;;
- : int list =
[2; 3; 5; 7; 11; 13; 17; 19; 23; 29; 31; 37; 41; 43; 47; 53; 59; 61; 67; 71;
73; 79; 83; 89; 97]
### Functional
```ocaml
(* first define some iterators *)
# let fold_iter f init a b =
let rec aux acc i =
if i > b
then (acc)
else aux (f acc i) (succ i)
in
aux init a ;;
val fold_iter : ('a -> int -> 'a) -> 'a -> int -> int -> 'a =
# let fold_step f init a b step =
let rec aux acc i =
if i > b
then (acc)
else aux (f acc i) (i + step)
in
aux init a ;;
val fold_step : ('a -> int -> 'a) -> 'a -> int -> int -> int -> 'a =
(* remove a given value from a list *)
# let remove li v =
let rec aux acc = function
| hd::tl when hd = v -> (List.rev_append acc tl)
| hd::tl -> aux (hd::acc) tl
| [] -> li
in
aux [] li ;;
val remove : 'a list -> 'a -> 'a list =
(* the main function *)
# let primes n =
let li =
(* create a list [from 2; ... until n] *)
List.rev(fold_iter (fun acc i -> (i::acc)) [] 2 n)
in
let limit = truncate(sqrt(float n)) in
fold_iter (fun li i ->
if List.mem i li (* test if (i) is prime *)
then (fold_step remove li (i*i) n i)
else li)
li 2 (pred limit)
;;
val primes : int -> int list =
# primes 200 ;;
- : int list =
[2; 3; 5; 7; 11; 13; 17; 19; 23; 29; 31; 37; 41; 43; 47; 53; 59; 61; 67; 71;
73; 79; 83; 89; 97; 101; 103; 107; 109; 113; 127; 131; 137; 139; 149; 151;
157; 163; 167; 173; 179; 181; 191; 193; 197; 199]
```
### Another functional version
This uses zero to denote struck-out numbers. It is slightly inefficient as it strikes-out multiples above p rather than p2
```ocaml
# let rec strike_nth k n l = match l with
| [] -> []
| h :: t ->
if k = 0 then 0 :: strike_nth (n-1) n t
else h :: strike_nth (k-1) n t;;
val strike_nth : int -> int -> int list -> int list =
# let primes n =
let limit = truncate(sqrt(float n)) in
let rec range a b = if a > b then [] else a :: range (a+1) b in
let rec sieve_primes l = match l with
| [] -> []
| 0 :: t -> sieve_primes t
| h :: t -> if h > limit then List.filter ((<) 0) l else
h :: sieve_primes (strike_nth (h-1) h t) in
sieve_primes (range 2 n) ;;
val primes : int -> int list =
# primes 200;;
- : int list =
[2; 3; 5; 7; 11; 13; 17; 19; 23; 29; 31; 37; 41; 43; 47; 53; 59; 61; 67; 71;
73; 79; 83; 89; 97; 101; 103; 107; 109; 113; 127; 131; 137; 139; 149; 151;
157; 163; 167; 173; 179; 181; 191; 193; 197; 199]
```
## Oforth
```Oforth
: eratosthenes(n)
| i j |
ListBuffer newSize(n) dup add(null) seqFrom(2, n) over addAll
2 n sqrt asInteger for: i [
dup at(i) ifNotNull: [ i sq n i step: j [ dup put(j, null) ] ]
]
filter(#notNull) ;
```
```txt
>100 eratosthenes println
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]
```
## Ol
```scheme
(define all (iota 999 2))
(print
(let main ((left '()) (right all))
(if (null? right)
(reverse left)
(unless (car right)
(main left (cdr right))
(let loop ((l '()) (r right) (n 0) (every (car right)))
(if (null? r)
(let ((l (reverse l)))
(main (cons (car l) left) (cdr l)))
(if (eq? n every)
(loop (cons #false l) (cdr r) 1 every)
(loop (cons (car r) l) (cdr r) (+ n 1) every)))))))
)
```
Output:
```txt
(2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659 661 673 677 683 691 701 709 719 727 733 739 743 751 757 761 769 773 787 797 809 811 821 823 827 829 839 853 857 859 863 877 881 883 887 907 911 919 929 937 941 947 953 967 971 977 983 991 997)
```
## Oz
```oz
declare
fun {Sieve N}
S = {Array.new 2 N true}
M = {Float.toInt {Sqrt {Int.toFloat N}}}
in
for I in 2..M do
if S.I then
for J in I*I..N;I do
S.J := false
end
end
end
S
end
fun {Primes N}
S = {Sieve N}
in
for I in 2..N collect:C do
if S.I then {C I} end
end
end
in
{Show {Primes 30}}
```
## PARI/GP
```parigp
Eratosthenes(lim)={
my(v=Vectorsmall(lim\1,unused,1));
forprime(p=2,sqrt(lim),
forstep(i=p^2,lim,p,
v[i]=0
)
);
for(i=1,lim,if(v[i],print1(i", ")))
};
```
An alternate version:
```parigp
Sieve(n)=
{
v=vector(n,unused,1);
for(i=2,sqrt(n),
if(v[i],
forstep(j=i^2,n,i,v[j]=0)));
for(i=2,n,if(v[i],print1(i)))
};
```
## Pascal
Note: Some Pascal implementations put quite low limits on the size of a set (e.g. Turbo Pascal doesn't allow more than 256 members). To compile on such an implementation, reduce the constant PrimeLimit accordingly.
```pascal
program primes(output)
const
PrimeLimit = 1000;
var
primes: set of 1 .. PrimeLimit;
n, k: integer;
needcomma: boolean;
begin
{ calculate the primes }
primes := [2 .. PrimeLimit];
for n := 1 to trunc(sqrt(PrimeLimit)) do
begin
if n in primes
then
begin
k := n*n;
while k < PrimeLimit do
begin
primes := primes - [k];
k := k + n
end
end
end;
{ output the primes }
needcomma := false;
for n := 1 to PrimeLimit do
if n in primes
then
begin
if needcomma
then
write(', ');
write(n);
needcomma := true
end
end.
```
### alternative using wheel
Using growing wheel to fill array for sieving for minimal unmark operations.
Sieving only with possible-prime factors.
```pascal
program prim(output);
//Sieve of Erathosthenes with fast elimination of multiples of small primes
{$IFNDEF FPC}
{$APPTYPE CONSOLE}
{$ENDIF}
const
PrimeLimit = 100*1000*1000;//1;
type
tLimit = 1..PrimeLimit;
var
//always initialized with 0 => false at startup
primes: array [tLimit] of boolean;
function BuildWheel: longInt;
//fill primfield with no multiples of small primes
//returns next sieveprime
//speedup ~1/3
var
//wheelprimes = 2,3,5,7,11... ;
//wheelsize = product [i= 0..wpno-1]wheelprimes[i] > Uint64 i> 13
wheelprimes :array[0..13] of byte;
wheelSize,wpno,
pr,pw,i, k: LongWord;
begin
//the mother of all numbers 1 ;-)
//the first wheel = generator of numbers
//not divisible by the small primes first found primes
pr := 1;
primes[1]:= true;
WheelSize := 1;
wpno := 0;
repeat
inc(pr);
//pw = pr projected in wheel of wheelsize
pw := pr;
if pw > wheelsize then
dec(pw,wheelsize);
If Primes[pw] then
begin
// writeln(pr:10,pw:10,wheelsize:16);
k := WheelSize+1;
//turn the wheel (pr-1)-times
for i := 1 to pr-1 do
begin
inc(k,WheelSize);
if k primeLimit then
k := primeLimit;
wheelPrimes[wpno] := pr;
primes[pr] := false;
inc(wpno);
//the new wheelsize
WheelSize := k;
//sieve multiples of the new found prime
i:= pr;
i := i*i;
while i <= k do
begin
primes[i] := false;
inc(i,pr);
end;
end;
until WheelSize >= PrimeLimit;
//re-insert wheel-primes
// 1 still stays prime
while wpno > 0 do
begin
dec(wpno);
primes[wheelPrimes[wpno]] := true;
end;
BuildWheel := pr+1;
end;
procedure Sieve;
var
sieveprime,
fakt : LongWord;
begin
//primes[1] = true is needed to stop for sieveprime = 2
// at //Search next smaller possible prime
sieveprime := BuildWheel;
//alternative here
//fillchar(primes,SizeOf(Primes),chr(ord(true)));sieveprime := 2;
repeat
if primes[sieveprime] then
begin
//eliminate 'possible prime' multiples of sieveprime
//must go downwards
//2*2 would unmark 4 -> 4*2 = 8 wouldnt be unmarked
fakt := PrimeLimit DIV sieveprime;
IF fakt < sieveprime then
BREAK;
repeat
//Unmark
primes[sieveprime*fakt] := false;
//Search next smaller possible prime
repeat
dec(fakt);
until primes[fakt];
until fakt < sieveprime;
end;
inc(sieveprime);
until false;
//remove 1
primes[1] := false;
end;
var
prCnt,
i : LongWord;
Begin
Sieve;
{count the primes }
prCnt := 0;
for i:= 1 to PrimeLimit do
inc(prCnt,Ord(primes[i]));
writeln(prCnt,' primes up to ',PrimeLimit);
end.
```
output: ( i3 4330 Haswell 3.5 Ghz fpc 2.6.4 -O3 )
```txt
5761455 primes up to 100000000
real 0m0.204s
user 0m0.193s
sys 0m0.013s
```
## Perl
For highest performance and ease, typically a module would be used, such as [https://metacpan.org/pod/Math::Prime::Util Math::Prime::Util], [https://metacpan.org/pod/Math::Prime::FastSieve Math::Prime::FastSieve], or [https://metacpan.org/pod/Math::Prime::XS Math::Prime::XS].
### Classic Sieve
```perl
sub sieve {
my $n = shift;
my @composite;
for my $i (2 .. int(sqrt($n))) {
if (!$composite[$i]) {
for (my $j = $i*$i; $j <= $n; $j += $i) {
$composite[$j] = 1;
}
}
}
my @primes;
for my $i (2 .. $n) {
$composite[$i] || push @primes, $i;
}
@primes;
}
```
===Odds only (faster)===
```perl
sub sieve2 {
my($n) = @_;
return @{([],[],[2],[2,3],[2,3])[$n]} if $n <= 4;
my @composite;
for (my $t = 3; $t*$t <= $n; $t += 2) {
if (!$composite[$t]) {
for (my $s = $t*$t; $s <= $n; $s += $t*2)
{ $composite[$s]++ }
}
}
my @primes = (2);
for (my $t = 3; $t <= $n; $t += 2) {
$composite[$t] || push @primes, $t;
}
@primes;
}
```
===Odds only, using vectors for lower memory use===
```perl
sub dj_vector {
my($end) = @_;
return @{([],[],[2],[2,3],[2,3])[$end]} if $end <= 4;
$end-- if ($end & 1) == 0; # Ensure end is odd
my ($sieve, $n, $limit, $s_end) = ( '', 3, int(sqrt($end)), $end >> 1 );
while ( $n <= $limit ) {
for (my $s = ($n*$n) >> 1; $s <= $s_end; $s += $n) {
vec($sieve, $s, 1) = 1;
}
do { $n += 2 } while vec($sieve, $n >> 1, 1) != 0;
}
my @primes = (2);
do { push @primes, 2*$_+1 if !vec($sieve,$_,1) } for (1..int(($end-1)/2));
@primes;
}
```
===Odds only, using strings for best performance===
Compared to array versions, about 2x faster (with 5.16.0 or later) and lower memory. Much faster than the experimental versions below. It's possible a mod-6 or mod-30 wheel could give more improvement, though possibly with obfuscation. The best next step for performance and functionality would be segmenting.
```perl
sub string_sieve {
my ($n, $i, $s, $d, @primes) = (shift, 7);
local $_ = '110010101110101110101110111110' .
'101111101110101110101110111110' x ($n/30);
until (($s = $i*$i) > $n) {
$d = $i<<1;
do { substr($_, $s, 1, '1') } until ($s += $d) > $n;
1 while substr($_, $i += 2, 1);
}
$_ = substr($_, 1, $n);
# For just the count: return ($_ =~ tr/0//);
push @primes, pos while m/0/g;
@primes;
}
```
This older version uses half the memory, but at the expense of a bit of speed and code complexity:
```perl
sub dj_string {
my($end) = @_;
return @{([],[],[2],[2,3],[2,3])[$end]} if $end <= 4;
$end-- if ($end & 1) == 0;
my $s_end = $end >> 1;
my $whole = int( ($end>>1) / 15); # prefill with 3 and 5 marked
my $sieve = '100010010010110' . '011010010010110' x $whole;
substr($sieve, ($end>>1)+1) = '';
my ($n, $limit, $s) = ( 7, int(sqrt($end)), 0 );
while ( $n <= $limit ) {
for ($s = ($n*$n) >> 1; $s <= $s_end; $s += $n) {
substr($sieve, $s, 1) = '1';
}
do { $n += 2 } while substr($sieve, $n>>1, 1);
}
# If you just want the count, it's very fast:
# my $count = 1 + $sieve =~ tr/0//;
my @primes = (2);
push @primes, 2*pos($sieve)-1 while $sieve =~ m/0/g;
@primes;
}
```
### Experimental
These are examples of golfing or unusual styles.
Golfing a bit, at the expense of speed:
```perl
sub sieve{ my (@s, $i);
grep { not $s[ $i = $_ ] and do
{ $s[ $i += $_ ]++ while $i <= $_[0]; 1 }
} 2..$_[0]
}
print join ", " => sieve 100;
```
Or with bit strings (much slower than the vector version above):
```perl
sub sieve{ my ($s, $i);
grep { not vec $s, $i = $_, 1 and do
{ (vec $s, $i += $_, 1) = 1 while $i <= $_[0]; 1 }
} 2..$_[0]
}
print join ", " => sieve 100;
```
A short recursive version:
```perl
sub erat {
my $p = shift;
return $p, $p**2 > $_[$#_] ? @_ : erat(grep $_%$p, @_)
}
print join ', ' => erat 2..100000;
```
Regexp (purely an example -- the regex engine limits it to only 32769):
```perl
sub sieve {
my ($s, $p) = "." . ("x" x shift);
1 while ++$p
and $s =~ /^(.{$p,}?)x/g
and $p = length($1)
and $s =~ s/(.{$p})./${1}./g
and substr($s, $p, 1) = "x";
$s
}
print sieve(1000);
```
### Extensible sieves
Here are two incremental versions, which allows one to create a tied array of primes:
```perl
use strict;
use warnings;
package Tie::SieveOfEratosthenes;
sub TIEARRAY {
my $class = shift;
bless \$class, $class;
}
# If set to true, produces copious output. Observing this output
# is an excellent way to gain insight into how the algorithm works.
use constant DEBUG => 0;
# If set to true, causes the code to skip over even numbers,
# improving runtime. It does not alter the output content, only the speed.
use constant WHEEL2 => 0;
BEGIN {
# This is loosely based on the Python implementation of this task,
# specifically the "Infinite generator with a faster algorithm"
my @primes = (2, 3);
my $ps = WHEEL2 ? 1 : 0;
my $p = $primes[$ps];
my $q = $p*$p;
my $incr = WHEEL2 ? 2 : 1;
my $candidate = $primes[-1] + $incr;
my %sieve;
print "Initial: p = $p, q = $q, candidate = $candidate\n" if DEBUG;
sub FETCH {
my $n = pop;
return if $n < 0;
return $primes[$n] if $n <= $#primes;
OUTER: while( 1 ) {
# each key in %sieve is a composite number between
# p and p-squared. Each value in %sieve is $incr x the prime
# which acted as a 'seed' for that key. We use the value
# to step through multiples of the seed-prime, until we find
# an empty slot in %sieve.
while( my $s = delete $sieve{$candidate} ) {
print "$candidate a multiple of ".($s/$incr).";\t\t" if DEBUG;
my $composite = $candidate + $s;
$composite += $s while exists $sieve{$composite};
print "The next stored multiple of ".($s/$incr)." is $composite\n" if DEBUG;
$sieve{$composite} = $s;
$candidate += $incr;
}
print "Candidate $candidate is not in sieve\n" if DEBUG;
while( $candidate < $q ) {
print "$candidate is prime\n" if DEBUG;
push @primes, $candidate;
$candidate += $incr;
next OUTER if exists $sieve{$candidate};
}
die "Candidate = $candidate, p = $p, q = $q" if $candidate > $q;
print "Candidate $candidate is equal to $p squared;\t" if DEBUG;
# Thus, it is now time to add p to the sieve,
my $step = $incr * $p;
my $composite = $q + $step;
$composite += $step while exists $sieve{$composite};
print "The next multiple of $p is $composite\n" if DEBUG;
$sieve{$composite} = $step;
# and fetch out a new value for p from our primes array.
$p = $primes[++$ps];
$q = $p * $p;
# And since $candidate was equal to some prime squared,
# it's obviously composite, and we need to increment it.
$candidate += $incr;
print "p is $p, q is $q, candidate is $candidate\n" if DEBUG;
} continue {
return $primes[$n] if $n <= $#primes;
}
}
}
if( !caller ) {
tie my (@prime_list), 'Tie::SieveOfEratosthenes';
my $limit = $ARGV[0] || 100;
my $line = "";
for( my $count = 0; $prime_list[$count] < $limit; ++$count ) {
$line .= $prime_list[$count]. ", ";
next if length($line) <= 70;
if( $line =~ tr/,// > 1 ) {
$line =~ s/^(.*,) (.*, )/$2/;
print $1, "\n";
} else {
print $line, "\n";
$line = "";
}
}
$line =~ s/, \z//;
print $line, "\n" if $line;
}
1;
```
This one is based on the vector sieve shown earlier, but adds to a list as needed, just sieving in the segment. Slightly faster and half the memory vs. the previous incremental sieve. It uses the same API -- arguably we should be offset by one so $primes[$n] returns the $n'th prime.
```perl
use strict;
use warnings;
package Tie::SieveOfEratosthenes;
sub TIEARRAY {
my $class = shift;
my @primes = (2,3,5,7);
return bless \@primes, $class;
}
sub prextend { # Extend the given list of primes using a segment sieve
my($primes, $to) = @_;
$to-- unless $to & 1; # Ensure end is odd
return if $to < $primes->[-1];
my $sqrtn = int(sqrt($to)+0.001);
prextend($primes, $sqrtn) if $primes->[-1] < $sqrtn;
my($segment, $startp) = ('', $primes->[-1]+1);
my($s_beg, $s_len) = ($startp >> 1, ($to>>1) - ($startp>>1));
for my $p (@$primes) {
last if $p > $sqrtn;
if ($p >= 3) {
my $p2 = $p*$p;
if ($p2 < $startp) { # Bump up to next odd multiple of p >= startp
my $f = 1+int(($startp-1)/$p);
$p2 = $p * ($f | 1);
}
for (my $s = ($p2>>1)-$s_beg; $s <= $s_len; $s += $p) {
vec($segment, $s, 1) = 1; # Mark composites in segment
}
}
}
# Now add all the primes found in the segment to the list
do { push @$primes, 1+2*($_+$s_beg) if !vec($segment,$_,1) } for 0 .. $s_len;
}
sub FETCHSIZE { 0x7FFF_FFFF } # Allows foreach to work
sub FETCH {
my($primes, $n) = @_;
return if $n < 0;
# Keep expanding the list as necessary, 5% larger each time.
prextend($primes, 1000+int(1.05*$primes->[-1])) while $n > $#$primes;
return $primes->[$n];
}
if( !caller ) {
tie my @prime_list, 'Tie::SieveOfEratosthenes';
my $limit = $ARGV[0] || 100;
print $prime_list[0];
my $i = 1;
while ($prime_list[$i] < $limit) { print " ", $prime_list[$i++]; }
print "\n";
}
1;
```
## Perl 6
```perl6
sub sieve( Int $limit ) {
my @is-prime = False, False, slip True xx $limit - 1;
gather for @is-prime.kv -> $number, $is-prime {
if $is-prime {
take $number;
loop (my $s = $number**2; $s <= $limit; $s += $number) {
@is-prime[$s] = False;
}
}
}
}
(sieve 100).join(",").say;
```
=== A set-based approach ===
More or less the same as the first Python example:
```perl6
sub eratsieve($n) {
# Requires n(1 - 1/(log(n-1))) storage
my $multiples = set();
gather for 2..$n -> $i {
unless $i (&) $multiples { # is subset
take $i;
$multiples (+)= set($i**2, *+$i ... (* > $n)); # union
}
}
}
say flat eratsieve(100);
```
This gives:
(2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97)
### Using a chain of filters
```perl6
sub primes ( UInt $n ) {
gather {
# create an iterator from 2 to $n (inclusive)
my $iterator := (2..$n).iterator;
loop {
# If it passed all of the filters it must be prime
my $prime := $iterator.pull-one;
# unless it is actually the end of the sequence
last if $prime =:= IterationEnd;
take $prime; # add the prime to the `gather` sequence
# filter out the factors of the current prime
$iterator := Seq.new($iterator).grep(* % $prime).iterator;
# (2..*).grep(* % 2).grep(* % 3).grep(* % 5).grep(* % 7)…
}
}
}
put primes( 100 );
```
Which prints
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
## Phix
```Phix
constant limit = 1000
sequence primes = {}
sequence flags = repeat(1, limit)
for i=2 to floor(sqrt(limit)) do
if flags[i] then
for k=i*i to limit by i do
flags[k] = 0
end for
end if
end for
for i=2 to limit do
if flags[i] then
primes &= i
end if
end for
? primes
```
```txt
{2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,
179,181,191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,293,307,311,313,317,331,337,347,349,353,359,367,
373,379,383,389,397,401,409,419,421,431,433,439,443,449,457,461,463,467,479,487,491,499,503,509,521,523,541,547,557,563,569,571,577,
587,593,599,601,607,613,617,619,631,641,643,647,653,659,661,673,677,683,691,701,709,719,727,733,739,743,751,757,761,769,773,787,797,
809,811,821,823,827,829,839,853,857,859,863,877,881,883,887,907,911,919,929,937,941,947,953,967,971,977,983,991,997}
```
See also [[Sexy_primes#Phix]] where the sieve is more useful than a list of primes, and [[Extensible_prime_generator#Phix]] for
a more memory efficient and therefore often faster and more appropriate method.
## PHP
```php
function iprimes_upto($limit)
{
for ($i = 2; $i < $limit; $i++)
{
$primes[$i] = true;
}
for ($n = 2; $n < $limit; $n++)
{
if ($primes[$n])
{
for ($i = $n*$n; $i < $limit; $i += $n)
{
$primes[$i] = false;
}
}
}
return $primes;
}
```
## PicoLisp
```PicoLisp
(de sieve (N)
(let Sieve (range 1 N)
(set Sieve)
(for I (cdr Sieve)
(when I
(for (S (nth Sieve (* I I)) S (nth (cdr S) I))
(set S) ) ) )
(filter bool Sieve) ) )
```
Output:
```txt
: (sieve 100)
-> (2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97)
```
## PL/I
```pli
eratos: proc options (main) reorder;
dcl i fixed bin (31);
dcl j fixed bin (31);
dcl n fixed bin (31);
dcl sn fixed bin (31);
dcl hbound builtin;
dcl sqrt builtin;
dcl sysin file;
dcl sysprint file;
get list (n);
sn = sqrt(n);
begin;
dcl primes(n) bit (1) aligned init ((*)((1)'1'b));
i = 2;
do while(i <= sn);
do j = i ** 2 by i to hbound(primes, 1);
/* Adding a test would just slow down processing! */
primes(j) = '0'b;
end;
do i = i + 1 to sn until(primes(i));
end;
end;
do i = 2 to hbound(primes, 1);
if primes(i) then
put data(i);
end;
end;
end eratos;
```
## Pony
```pony
use "time" // for testing
use "collections"
class Primes is Iterator[U32] // returns an Iterator of found primes...
let _bitmask: Array[U8] = [ 1; 2; 4; 8; 16; 32; 64; 128 ]
var _lmt: USize
let _cmpsts: Array[U8]
var _ndx: USize = 2
var _curr: U32 = 2
new create(limit: U32) ? =>
_lmt = USize.from[U32](limit)
let sqrtlmt = USize.from[F64](F64.from[U32](limit).sqrt())
_cmpsts = Array[U8].init(0, (_lmt + 8) / 8) // already zeroed; bit array
_cmpsts(0)? = 3 // mark 0 and 1 as not prime!
if sqrtlmt < 2 then return end
for p in Range[USize](2, sqrtlmt + 1) do
if (_cmpsts(p >> 3)? and _bitmask(p and 7)?) == 0 then
var s = p * p // cull start address for p * p!
let slmt = (s + (p << 3)).min(_lmt + 1)
while s < slmt do
let msk = _bitmask(s and 7)?
var c = s >> 3
while c < _cmpsts.size() do
_cmpsts(c)? = _cmpsts(c)? or msk
c = c + p
end
s = s + p
end
end
end
fun ref has_next(): Bool val => _ndx < (_lmt + 1)
fun ref next(): U32 ? =>
_curr = U32.from[USize](_ndx); _ndx = _ndx + 1
while (_ndx <= _lmt) and ((_cmpsts(_ndx >> 3)? and _bitmask(_ndx and 7)?) != 0) do
_ndx = _ndx + 1
end
_curr
actor Main
new create(env: Env) =>
let limit: U32 = 1_000_000_000
try
env.out.write("Primes to 100: ")
for p in Primes(100)? do env.out.write(p.string() + " ") end
var count: I32 = 0
for p in Primes(1_000_000)? do count = count + 1 end
env.out.print("\nThere are " + count.string() + " primes to a million.")
let t = Time
let start = t.millis()
let prms = Primes(limit)?
let elpsd = t.millis() - start
count = 0
for _ in prms do count = count + 1 end
env.out.print("Found " + count.string() + " primes to " + limit.string() + ".")
env.out.print("This took " + elpsd.string() + " milliseconds.")
end
```
```txt
Primes to 100: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
There are 78498 primes to a million.
Found 50847534 primes to 1000000000.
This took 28123 milliseconds.
```
Note to users: a naive monolithic sieve (one huge array) isn't really the way to implement this for other than trivial usage in sieving ranges to a few millions as cache locality becomes a very large problem as the size of the array (even bit packed with one bit per number representation as here) limits the maximum range that can be sieved and the "cache thrashing" limits the speed.
For extended ranges, a Page Segmented version should be used. As well, for any extended ranges in the billions, it is a waste of available computer resources to not use the multi-threading available in a modern CPU, at which Pony would do very well with its built-in Actor concurrency model.
These versions use "loop unpeeling" (not full loop unrolling), which recognizes the repeating modulo pattern of masking the bytes by the base primes less than the square root of the limit so that an "unpeeling" by eight loops can cull by a constant bit mask over the whole range. For smaller ranges where the speed is not limited by "cache thrashing", this can provide about a factor-of-two speed-up.
===Alternate Odds-Only version of the above===
It is a waste not to do the trivial changes to the above code to sieve odds-only, which is about two and a half times faster due to the decreased number of culling operations; it doesn't really do much about the huge array problem though, other than to reduce it by a factor of two.
```pony
use "time" // for testing
use "collections"
class Primes is Iterator[U32] // returns an Iterator of found primes...
let _bitmask: Array[U8] = [ 1; 2; 4; 8; 16; 32; 64; 128 ]
var _lmti: USize
let _cmpsts: Array[U8]
var _ndx: USize = 0
var _curr: U32 = 0
new create(limit: U32) ? =>
if limit < 3 then _lmti = 0; _cmpsts = Array[U8](); return end
_lmti = USize.from[U32]((limit - 3) / 2)
let sqrtlmti = (USize.from[F64](F64.from[U32](limit).sqrt()) - 3) / 2
_cmpsts = Array[U8].init(0, (_lmti + 8) / 8) // already zeroed; bit array
for i in Range[USize](0, sqrtlmti + 1) do
if (_cmpsts(i >> 3)? and _bitmask(i and 7)?) == 0 then
let p = i + i + 3
var s = ((i << 1) * (i + 3)) + 3 // cull start address for p * p!
let slmt = (s + (p << 3)).min(_lmti + 1)
while s < slmt do
let msk = _bitmask(s and 7)?
var c = s >> 3
while c < _cmpsts.size() do
_cmpsts(c)? = _cmpsts(c)? or msk
c = c + p
end
s = s + p
end
end
end
fun ref has_next(): Bool val => _ndx < (_lmti + 1)
fun ref next(): U32 ? =>
if _curr < 1 then _curr = 3; if _lmti == 0 then _ndx = 1 end; return 2 end
_curr = U32.from[USize](_ndx + _ndx + 3); _ndx = _ndx + 1
while (_ndx <= _lmti) and ((_cmpsts(_ndx >> 3)? and _bitmask(_ndx and 7)?) != 0) do
_ndx = _ndx + 1
end
_curr
actor Main
new create(env: Env) =>
let limit: U32 = 1_000_000_000
try
env.out.write("Primes to 100: ")
for p in Primes(100)? do env.out.write(p.string() + " ") end
var count: I32 = 0
for p in Primes(1_000_000)? do count = count + 1 end
env.out.print("\nThere are " + count.string() + " primes to a million.")
let t = Time
let start = t.millis()
let prms = Primes(limit)?
let elpsd = t.millis() - start
count = 0
for _ in prms do count = count + 1 end
env.out.print("Found " + count.string() + " primes to " + limit.string() + ".")
env.out.print("This took " + elpsd.string() + " milliseconds.")
end
```
The output is the same as the above except that it is about two and a half times faster due to that many less culling operations.
## Pop11
```txt
define eratostenes(n);
lvars bits = inits(n), i, j;
for i from 2 to n do
if bits(i) = 0 then
printf('' >< i, '%s\n');
for j from 2*i by i to n do
1 -> bits(j);
endfor;
endif;
endfor;
enddefine;
```
## PowerShell
### Basic procedure
It outputs immediately so that the number can be used by the pipeline.
```PowerShell
function Sieve ( [int] $num )
{
$isprime = @{}
2..$num | Where-Object {
$isprime[$_] -eq $null } | ForEach-Object {
$_
$isprime[$_] = $true
$i=$_*$_
for ( ; $i -le $num; $i += $_ )
{ $isprime[$i] = $false }
}
}
```
### Another implementation
```PowerShell
function eratosthenes ($n) {
if($n -ge 1){
$prime = @(1..($n+1) | foreach{$true})
$prime[1] = $false
$m = [Math]::Floor([Math]::Sqrt($n))
for($i = 2; $i -le $m; $i++) {
if($prime[$i]) {
for($j = $i*$i; $j -le $n; $j += $i) {
$prime[$j] = $false
}
}
}
1..$n | where{$prime[$_]}
} else {
Write-Warning "$n is less than 1"
}
}
"$(eratosthenes 100)"
```
Output:
```txt
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
```
## Processing
Calculate the primes up to 1000000 with Processing, including a visualisation of the process. As an additional visual effect, the layout of the pixel could be changed from the line-by-line layout to a spiral-like layout starting in the middle of the screen.
```java
int maxx,maxy;
int max;
boolean[] sieve;
void plot(int pos, boolean active) {
set(pos%maxx,pos/maxx, active?#000000:#ffffff);
}
void setup() {
size(1000, 1000, P2D);
frameRate(2);
maxx=width;
maxy=height;
max=width*height;
sieve=new boolean[max+1];
sieve[1]=false;
plot(0,false);
plot(1,false);
for(int i=2;i<=max;i++) {
sieve[i]=true;
plot(i,true);
}
}
int i=2;
void draw() {
if(!sieve[i]) {
while(i*i R = [H1|R1], filter(H, H2, T, R1)
; H3 is H2 + H,
( H1 =:= H2 -> filter(H, H3, T, R)
; filter(H, H3, [H1|T], R) ) ).
```
```txt
?- time(( primes(7920,X), length(X,N) )).
% 1,131,127 inferences, 0.109 CPU in 0.125 seconds (88% CPU, 10358239 Lips)
X = [2, 3, 5, 7, 11, 13, 17, 19, 23|...],
N = 1000 .
```
====Basic bounded Euler's sieve====
This is actually the Euler's variant of the sieve of Eratosthenes, generating (and thus removing) each multiple only once, though a sub-optimal implementation.
```Prolog
primes(X, PS) :- X > 1, range(2, X, R), sieve(R, PS).
range(X, X, [X]) :- !.
range(X, Y, [X | R]) :- X < Y, X1 is X + 1, range(X1, Y, R).
mult(A, B, C) :- C is A*B.
sieve([X], [X]) :- !.
sieve([H | T], [H | S]) :- maplist( mult(H), [H | T], MS),
remove(MS, T, R), sieve(R, S).
remove( _, [], [] ) :- !.
remove( [H | X], [H | Y], R ) :- !, remove(X, Y, R).
remove( X, [H | Y], [H | R]) :- remove(X, Y, R).
```
Running in SWI Prolog,
```txt
?- time(( primes(7920,X), length(X,N) )).
% 2,087,373 inferences, 0.203 CPU in 0.203 seconds (100% CPU, 10297621 Lips)
X = [2, 3, 5, 7, 11, 13, 17, 19, 23|...],
N = 1000.
```
====Optimized Euler's sieve====
We can stop early, with massive improvement in complexity (below ~ n1.5 inferences, empirically, vs. the ~ n2 of the above, in ''n'' primes produced; showing only the modified predicates):
```Prolog
primes(X, PS) :- X > 1, range(2, X, R), sieve(X, R, PS).
sieve(X, [H | T], [H | T]) :- H*H > X, !.
sieve(X, [H | T], [H | S]) :- maplist( mult(H), [H | T], MS),
remove(MS, T, R), sieve(X, R, S).
```
```txt
?- time(( primes(7920,X), length(X,N) )).
% 174,437 inferences, 0.016 CPU in 0.016 seconds (100% CPU, 11181787 Lips)
X = [2, 3, 5, 7, 11, 13, 17, 19, 23|...],
N = 1000.
```
### =Bounded sieve=
Optimized by stopping early, traditional sieve of Eratosthenes generating multiples by iterated addition.
```Prolog
primes(X, PS) :- X > 1, range(2, X, R), sieve(X, R, PS).
range(X, X, [X]) :- !.
range(X, Y, [X | R]) :- X < Y, X1 is X + 1, range(X1, Y, R).
sieve(X, [H | T], [H | T]) :- H*H > X, !.
sieve(X, [H | T], [H | S]) :- mults( H, X, MS), remove(MS, T, R), sieve(X, R, S).
mults( H, Lim, MS):- M is H*H, mults( H, M, Lim, MS).
mults( _, M, Lim, []):- M > Lim, !.
mults( H, M, Lim, [M|MS]):- M2 is M+H, mults( H, M2, Lim, MS).
remove( _, [], [] ) :- !.
remove( [H | X], [H | Y], R ) :- !, remove(X, Y, R).
remove( [H | X], [G | Y], R ) :- H < G, !, remove(X, [G | Y], R).
remove( X, [H | Y], [H | R]) :- remove(X, Y, R).
```
```txt
?- time(( primes(7920,X), length(X,N) )).
% 140,654 inferences, 0.016 CPU in 0.011 seconds (142% CPU, 9016224 Lips)
X = [2, 3, 5, 7, 11, 13, 17, 19, 23|...],
N = 1000.
```
### Using lazy lists
In SWI Prolog and others, where freeze/2 is available.
### =Basic variant=
```prolog
primes(PS):- count(2, 1, NS), sieve(NS, PS).
count(N, D, [N|T]):- freeze(T, (N2 is N+D, count(N2, D, T))).
sieve([N|NS],[N|PS]):- N2 is N*N, count(N2,N,A), remove(A,NS,B), freeze(PS, sieve(B,PS)).
take(N, X, A):- length(A, N), append(A, _, X).
remove([A|T],[B|S],R):- A < B -> remove(T,[B|S],R) ;
A=:=B -> remove(T,S,R) ;
R = [B|R2], freeze(R2, remove([A|T], S, R2)).
```
```txt
?- time(( primes(PS), take(1000,PS,R1), length(R,10), append(_,R,R1), writeln(R), false )).
[7841,7853,7867,7873,7877,7879,7883,7901,7907,7919]
% 8,464,518 inferences, 0.702 CPU in 0.697 seconds (101% CPU, 12057641 Lips)
false.
```
### =Optimized by postponed removal=
Showing only changed predicates.
```prolog
primes([2|PS]):-
freeze(PS, (primes(BPS), count(3, 1, NS), sieve(NS, BPS, 4, PS))).
sieve([N|NS], BPS, Q, PS):-
N < Q -> PS = [N|PS2], freeze(PS2, sieve(NS, BPS, Q, PS2))
; BPS = [BP,BP2|BPS2], Q2 is BP2*BP2, count(Q, BP, MS),
remove(MS, NS, R), sieve(R, [BP2|BPS2], Q2, PS).
```
```txt
?- time(( primes(PS), take(1000,PS,R1), length(R,10), append(_,R,R1), writeln(R), false )).
[7841,7853,7867,7873,7877,7879,7883,7901,7907,7919]
% 697,727 inferences, 0.078 CPU in 0.078 seconds (100% CPU, 8945161 Lips)
false. %% odds only: 487,441 inferences
```
### Using facts to record composite numbers
The first two solutions use Prolog "facts" to record the composite (i.e. already-visited) numbers.
==== Elementary approach: multiplication-free, division-free, mod-free, and cut-free====
The basic Eratosthenes sieve depends on nothing more complex than counting.
In celebration of this simplicity, the first approach to the problem taken here is
free of multiplication and division, as well as Prolog's non-logical "cut".
It defines the predicate between/4 to avoid division, and composite/1
to record integers that are found to be composite.
```Prolog
% %sieve( +N, -Primes ) is true if Primes is the list of consecutive primes
% that are less than or equal to N
sieve( N, [2|Rest]) :-
retractall( composite(_) ),
sieve( N, 2, Rest ) -> true. % only one solution
% sieve P, find the next non-prime, and then recurse:
sieve( N, P, [I|Rest] ) :-
sieve_once(P, N),
(P = 2 -> P2 is P+1; P2 is P+2),
between(P2, N, I),
(composite(I) -> fail; sieve( N, I, Rest )).
% It is OK if there are no more primes less than or equal to N:
sieve( N, P, [] ).
sieve_once(P, N) :-
forall( between(P, N, P, IP),
(composite(IP) -> true ; assertz( composite(IP) )) ).
% To avoid division, we use the iterator
% between(+Min, +Max, +By, -I)
% where we assume that By > 0
% This is like "for(I=Min; I <= Max; I+=By)" in C.
between(Min, Max, By, I) :-
Min =< Max,
A is Min + By,
(I = Min; between(A, Max, By, I) ).
% Some Prolog implementations require the dynamic predicates be
% declared:
:- dynamic( composite/1 ).
```
The above has been tested with SWI-Prolog and gprolog.
```Prolog
% SWI-Prolog:
?- time( (sieve(100000,P), length(P,N), writeln(N), last(P, LP), writeln(LP) )).
% 1,323,159 inferences, 0.862 CPU in 0.921 seconds (94% CPU, 1534724 Lips)
P = [2, 3, 5, 7, 11, 13, 17, 19, 23|...],
N = 9592,
LP = 99991.
```
### = Optimized approach=
[http://ideone.com/WDC7z Works with SWI-Prolog].
```Prolog
sieve(N, [2|PS]) :- % PS is list of odd primes up to N
retractall(mult(_)),
sieve_O(3,N,PS).
sieve_O(I,N,PS) :- % sieve odds from I up to N to get PS
I =< N, !, I1 is I+2,
( mult(I) -> sieve_O(I1,N,PS)
; ( I =< N / I ->
ISq is I*I, DI is 2*I, add_mults(DI,ISq,N)
; true
),
PS = [I|T],
sieve_O(I1,N,T)
).
sieve_O(I,N,[]) :- I > N.
add_mults(DI,I,N) :-
I =< N, !,
( mult(I) -> true ; assert(mult(I)) ),
I1 is I+DI,
add_mults(DI,I1,N).
add_mults(_,I,N) :- I > N.
main(N) :- current_prolog_flag(verbose,F),
set_prolog_flag(verbose,normal),
time( sieve( N,P)), length(P,Len), last(P, LP), writeln([Len,LP]),
set_prolog_flag(verbose,F).
:- dynamic( mult/1 ).
:- main(100000), main(1000000).
```
Running it produces
```Prolog
%% stdout copy
[9592, 99991]
[78498, 999983]
%% stderr copy
% 293,176 inferences, 0.14 CPU in 0.14 seconds (101% CPU, 2094114 Lips)
% 3,122,303 inferences, 1.63 CPU in 1.67 seconds (97% CPU, 1915523 Lips)
```
which indicates ~ N1.1 [http://en.wikipedia.org/wiki/Analysis_of_algorithms#Empirical_orders_of_growth empirical orders of growth], which is consistent with the ''O(N log log N)'' theoretical runtime complexity.
### Using a priority queue
Uses a ariority queue, from the paper "The Genuine Sieve of Eratosthenes" by Melissa O'Neill. Works with YAP (Yet Another Prolog)
```Prolog
?- use_module(library(heaps)).
prime(2).
prime(N) :- prime_heap(N, _).
prime_heap(3, H) :- list_to_heap([9-6], H).
prime_heap(N, H) :-
prime_heap(M, H0), N0 is M + 2,
next_prime(N0, H0, N, H).
next_prime(N0, H0, N, H) :-
\+ min_of_heap(H0, N0, _),
N = N0, Composite is N*N, Skip is N+N,
add_to_heap(H0, Composite, Skip, H).
next_prime(N0, H0, N, H) :-
min_of_heap(H0, N0, _),
adjust_heap(H0, N0, H1), N1 is N0 + 2,
next_prime(N1, H1, N, H).
adjust_heap(H0, N, H) :-
min_of_heap(H0, N, _),
get_from_heap(H0, N, Skip, H1),
Composite is N + Skip, add_to_heap(H1, Composite, Skip, H2),
adjust_heap(H2, N, H).
adjust_heap(H, N, H) :-
\+ min_of_heap(H, N, _).
```
## PureBasic
### Basic procedure
```PureBasic
For n=2 To Sqr(lim)
If Nums(n)=0
m=n*n
While m<=lim
Nums(m)=1
m+n
Wend
EndIf
Next n
```
### Working example
```PureBasic
Dim Nums.i(0)
Define l, n, m, lim
If OpenConsole()
; Ask for the limit to search, get that input and allocate a Array
Print("Enter limit for this search: ")
lim=Val(Input())
ReDim Nums(lim)
; Use a basic Sieve of Eratosthenes
For n=2 To Sqr(lim)
If Nums(n)=#False
m=n*n
While m<=lim
Nums(m)=#True
m+n
Wend
EndIf
Next n
;Present the result to our user
PrintN(#CRLF$+"The Prims up to "+Str(lim)+" are;")
m=0: l=Log10(lim)+1
For n=2 To lim
If Nums(n)=#False
Print(RSet(Str(n),l)+" ")
m+1
If m>72/(l+1)
m=0: PrintN("")
EndIf
EndIf
Next
Print(#CRLF$+#CRLF$+"Press ENTER to exit"): Input()
CloseConsole()
EndIf
```
Output may look like;
Enter limit for this search: 750
The Prims up to 750 are;
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
53 59 61 67 71 73 79 83 89 97 101 103 107 109 113
127 131 137 139 149 151 157 163 167 173 179 181 191 193 197
199 211 223 227 229 233 239 241 251 257 263 269 271 277 281
283 293 307 311 313 317 331 337 347 349 353 359 367 373 379
383 389 397 401 409 419 421 431 433 439 443 449 457 461 463
467 479 487 491 499 503 509 521 523 541 547 557 563 569 571
577 587 593 599 601 607 613 617 619 631 641 643 647 653 659
661 673 677 683 691 701 709 719 727 733 739 743
Press ENTER to exit
## Python
Note that the examples use range instead of xrange for Python 3 and Python 2 compatability, but when using Python 2 xrange is the nearest equivalent to Python 3's implementation of range and should be substituted for ranges with a very large number of items.
### Using set lookup
The version below uses a set to store the multiples. set objects are much faster (usually O(log n)) than lists (O(n)) for checking if a given object is a member. Using the set.update method
avoids explicit iteration in the interpreter, giving a further speed improvement.
```python
def eratosthenes2(n):
multiples = set()
for i in range(2, n+1):
if i not in multiples:
yield i
multiples.update(range(i*i, n+1, i))
print(list(eratosthenes2(100)))
```
### Using array lookup
The version below uses array lookup to test for primality. The function primes_upto() is a straightforward implementation of [http://en.wikipedia.org/wiki/Sieve_of_Eratosthenes#Algorithm Sieve of Eratosthenes]algorithm. It returns prime numbers less than or equal to limit.
```python
def primes_upto(limit):
is_prime = [False] * 2 + [True] * (limit - 1)
for n in range(int(limit**0.5 + 1.5)): # stop at ``sqrt(limit)``
if is_prime[n]:
for i in range(n*n, limit+1, n):
is_prime[i] = False
return [i for i, prime in enumerate(is_prime) if prime]
```
### Using generator
The following code may be slightly slower than using the array/list as above, but uses no memory for output:
```python
def iprimes_upto(limit):
is_prime = [False] * 2 + [True] * (limit - 1)
for n in xrange(int(limit**0.5 + 1.5)): # stop at ``sqrt(limit)``
if is_prime[n]:
for i in range(n * n, limit + 1, n): # start at ``n`` squared
is_prime[i] = False
for i in xrange(limit + 1):
if is_prime[i]: yield i
```
```python>>>
list(iprimes_upto(15))
[2, 3, 5, 7, 11, 13]
```
===Odds-only version of the array sieve above===
The following code is faster than the above array version using only odd composite operations (for a factor of over two) and because it has been optimized to use slice operations for composite number culling to avoid extra work by the interpreter:
```python
def primes2(limit):
if limit < 2: return []
if limit < 3: return [2]
lmtbf = (limit - 3) // 2
buf = [True] * (lmtbf + 1)
for i in range((int(limit ** 0.5) - 3) // 2 + 1):
if buf[i]:
p = i + i + 3
s = p * (i + 1) + i
buf[s::p] = [False] * ((lmtbf - s) // p + 1)
return [2] + [i + i + 3 for i, v in enumerate(buf) if v]
```
Note that "range" needs to be changed to "xrange" for maximum speed with Python 2.
===Odds-only version of the generator version above===
The following code is faster than the above generator version using only odd composite operations (for a factor of over two) and because it has been optimized to use slice operations for composite number culling to avoid extra work by the interpreter:
```python
def iprimes2(limit):
yield 2
if limit < 3: return
lmtbf = (limit - 3) // 2
buf = [True] * (lmtbf + 1)
for i in range((int(limit ** 0.5) - 3) // 2 + 1):
if buf[i]:
p = i + i + 3
s = p * (i + 1) + i
buf[s::p] = [False] * ((lmtbf - s) // p + 1)
for i in range(lmtbf + 1):
if buf[i]: yield (i + i + 3)
```
Note that this version may actually run slightly faster than the equivalent array version with the advantage that the output doesn't require any memory.
Also note that "range" needs to be changed to "xrange" for maximum speed with Python 2.
### Factorization wheel235 version of the generator version
This uses a 235 factorial wheel for further reductions in operations; the same techniques can be applied to the array version as well; it runs slightly faster and uses slightly less memory as compared to the odds-only algorithms:
```python
def primes235(limit):
yield 2; yield 3; yield 5
if limit < 7: return
modPrms = [7,11,13,17,19,23,29,31]
gaps = [4,2,4,2,4,6,2,6,4,2,4,2,4,6,2,6] # 2 loops for overflow
ndxs = [0,0,0,0,1,1,2,2,2,2,3,3,4,4,4,4,5,5,5,5,5,5,6,6,7,7,7,7,7,7]
lmtbf = (limit + 23) // 30 * 8 - 1 # integral number of wheels rounded up
lmtsqrt = (int(limit ** 0.5) - 7)
lmtsqrt = lmtsqrt // 30 * 8 + ndxs[lmtsqrt % 30] # round down on the wheel
buf = [True] * (lmtbf + 1)
for i in range(lmtsqrt + 1):
if buf[i]:
ci = i & 7; p = 30 * (i >> 3) + modPrms[ci]
s = p * p - 7; p8 = p << 3
for j in range(8):
c = s // 30 * 8 + ndxs[s % 30]
buf[c::p8] = [False] * ((lmtbf - c) // p8 + 1)
s += p * gaps[ci]; ci += 1
for i in range(lmtbf - 6 + (ndxs[(limit - 7) % 30])): # adjust for extras
if buf[i]: yield (30 * (i >> 3) + modPrms[i & 7])
```
Note: Much of the time (almost two thirds for this last case for Python 2.7.6) for any of these array/list or generator algorithms is used in the computation and enumeration of the final output in the last line(s), so any slight changes to those lines can greatly affect execution time. For Python 3 this enumeration is about twice as slow as Python 2 (Python 3.3 slow and 3.4 slower) for an even bigger percentage of time spent just outputting the results. This slow enumeration means that there is little advantage to versions that use even further wheel factorization, as the composite number culling is a small part of the time to enumerate the results.
If just the count of the number of primes over a range is desired, then converting the functions to prime counting functions by changing the final enumeration lines to "return buf.count(True)" will save a lot of time.
Note that "range" needs to be changed to "xrange" for maximum speed with Python 2 where Python 2's "xrange" is a better choice for very large sieve ranges.
Timings were done primarily in Python 2 although source is Python 2/3 compatible (shows range and not xrange).
### Using numpy
Below code adapted from [http://en.literateprograms.org/Sieve_of_Eratosthenes_(Python,_arrays)#simple_implementation literateprograms.org] using [http://numpy.scipy.org/ numpy]
```python
import numpy
def primes_upto2(limit):
is_prime = numpy.ones(limit + 1, dtype=numpy.bool)
for n in xrange(2, int(limit**0.5 + 1.5)):
if is_prime[n]:
is_prime[n*n::n] = 0
return numpy.nonzero(is_prime)[0][2:]
```
'''Performance note:''' there is no point to add wheels here, due to execution of p[n*n::n] = 0 and nonzero() takes us almost all time.
Also see [http://rebrained.com/?p=458 Prime numbers and Numpy – Python].
### Using wheels with numpy
Version with wheel based optimization:
```python
from numpy import array, bool_, multiply, nonzero, ones, put, resize
#
def makepattern(smallprimes):
pattern = ones(multiply.reduce(smallprimes), dtype=bool_)
pattern[0] = 0
for p in smallprimes:
pattern[p::p] = 0
return pattern
#
def primes_upto3(limit, smallprimes=(2,3,5,7,11)):
sp = array(smallprimes)
if limit <= sp.max(): return sp[sp <= limit]
#
isprime = resize(makepattern(sp), limit + 1)
isprime[:2] = 0; put(isprime, sp, 1)
#
for n in range(sp.max() + 2, int(limit**0.5 + 1.5), 2):
if isprime[n]:
isprime[n*n::n] = 0
return nonzero(isprime)[0]
```
Examples:
```python>>>
primes_upto3(10**6, smallprimes=(2,3)) # Wall time: 0.17
array([ 2, 3, 5, ..., 999961, 999979, 999983])
>>> primes_upto3(10**7, smallprimes=(2,3)) # Wall time: '''2.13'''
array([ 2, 3, 5, ..., 9999971, 9999973, 9999991])
>>> primes_upto3(15)
array([ 2, 3, 5, 7, 11, 13])
>>> primes_upto3(10**7, smallprimes=primes_upto3(15)) # Wall time: '''1.31'''
array([ 2, 3, 5, ..., 9999971, 9999973, 9999991])
>>> primes_upto2(10**7) # Wall time: '''1.39''' <-- version ''without'' wheels
array([ 2, 3, 5, ..., 9999971, 9999973, 9999991])
>>> primes_upto3(10**7) # Wall time: '''1.30'''
array([ 2, 3, 5, ..., 9999971, 9999973, 9999991])
```
The above-mentioned examples demonstrate that the ''given'' wheel based optimization does not show significant performance gain.
### Infinite generator
A generator that will generate primes indefinitely (perhaps until it runs out of memory). Used as a library [[Extensible prime generator#Python|here]].
```python
import heapq
# generates all prime numbers
def sieve():
# priority queue of the sequences of non-primes
# the priority queue allows us to get the "next" non-prime quickly
nonprimes = []
i = 2
while True:
if nonprimes and i == nonprimes[0][0]: # non-prime
while nonprimes[0][0] == i:
# for each sequence that generates this number,
# have it go to the next number (simply add the prime)
# and re-position it in the priority queue
x = nonprimes[0]
x[0] += x[1]
heapq.heapreplace(nonprimes, x)
else: # prime
# insert a 2-element list into the priority queue:
# [current multiple, prime]
# the first element allows sorting by value of current multiple
# we start with i^2
heapq.heappush(nonprimes, [i*i, i])
yield i
i += 1
```
Example:
```txt
>>> foo = sieve()
>>> for i in range(8):
... print(next(foo))
...
2
3
5
7
11
13
17
19
```
### Infinite generator with a faster algorithm
The adding of each discovered prime's incremental step info to the mapping should be '''''postponed''''' until the prime's ''square'' is seen amongst the candidate numbers, as it is useless before that point. This drastically reduces the space complexity from O(n) to O(sqrt(n/log(n))), in ''n'' primes produced, and also lowers the run time complexity quite low ([http://ideone.com/VXep9F this test entry in Python 2.7] and [http://ideone.com/muuS4H this test entry in Python 3.x] shows about ~ n1.08 [http://en.wikipedia.org/wiki/Analysis_of_algorithms#Empirical_orders_of_growth empirical order of growth] which is very close to the theoretical value of O(n log(n) log(log(n))), in ''n'' primes produced):
```python
def primes():
yield 2; yield 3; yield 5; yield 7;
bps = (p for p in primes()) # separate supply of "base" primes (b.p.)
p = next(bps) and next(bps) # discard 2, then get 3
q = p * p # 9 - square of next base prime to keep track of,
sieve = {} # in the sieve dict
n = 9 # n is the next candidate number
while True:
if n not in sieve: # n is not a multiple of any of base primes,
if n < q: # below next base prime's square, so
yield n # n is prime
else:
p2 = p + p # n == p * p: for prime p, add p * p + 2 * p
sieve[q + p2] = p2 # to the dict, with 2 * p as the increment step
p = next(bps); q = p * p # pull next base prime, and get its square
else:
s = sieve.pop(n); nxt = n + s # n's a multiple of some b.p., find next multiple
while nxt in sieve: nxt += s # ensure each entry is unique
sieve[nxt] = s # nxt is next non-marked multiple of this prime
n += 2 # work on odds only
import itertools
def primes_up_to(limit):
return list(itertools.takewhile(lambda p: p <= limit, primes()))
```
### Fast infinite generator using a wheel
Although theoretically over three times faster than odds-only, the following code using a 2/3/5/7 wheel is only about 1.5 times faster than the above odds-only code due to the extra overheads in code complexity. The [http://ideone.com/LFaRnT test link for Python 2.7] and [http://ideone.com/ZAY0T2 test link for Python 3.x] show about the same empirical order of growth as the odds-only implementation above once the range grows enough so the dict operations become amortized to a constant factor.
```python
def primes():
for p in [2,3,5,7]: yield p # base wheel primes
gaps1 = [ 2,4,2,4,6,2,6,4,2,4,6,6,2,6,4,2,6,4,6,8,4,2,4,2,4,8 ]
gaps = gaps1 + [ 6,4,6,2,4,6,2,6,6,4,2,4,6,2,6,4,2,4,2,10,2,10 ] # wheel2357
def wheel_prime_pairs():
yield (11,0); bps = wheel_prime_pairs() # additional primes supply
p, pi = next(bps); q = p * p # adv to get 11 sqr'd is 121 as next square to put
sieve = {}; n = 13; ni = 1 # into sieve dict; init cndidate, wheel ndx
while True:
if n not in sieve: # is not a multiple of previously recorded primes
if n < q: yield (n, ni) # n is prime with wheel modulo index
else:
npi = pi + 1 # advance wheel index
if npi > 47: npi = 0
sieve[q + p * gaps[pi]] = (p, npi) # n == p * p: put next cull position on wheel
p, pi = next(bps); q = p * p # advance next prime and prime square to put
else:
s, si = sieve.pop(n)
nxt = n + s * gaps[si] # move current cull position up the wheel
si = si + 1 # advance wheel index
if si > 47: si = 0
while nxt in sieve: # ensure each entry is unique by wheel
nxt += s * gaps[si]
si = si + 1 # advance wheel index
if si > 47: si = 0
sieve[nxt] = (s, si) # next non-marked multiple of a prime
nni = ni + 1 # advance wheel index
if nni > 47: nni = 0
n += gaps[ni]; ni = nni # advance on the wheel
for p, pi in wheel_prime_pairs(): yield p # strip out indexes
```
Further gains of about 1.5 times in speed can be made using the same code by only changing the tables and a few constants for a further constant factor gain of about 1.5 times in speed by using a 2/3/5/7/11/13/17 wheel (with the gaps list 92160 elements long) computed for a slight constant overhead time as per the [http://ideone.com/4Ld26g test link for Python 2.7] and [http://ideone.com/72Dmyt test link for Python 3.x]. Further wheel factorization will not really be worth it as the gains will be small (if any and not losses) and the gaps table huge - it is already too big for efficient use by 32-bit Python 3 and the wheel should likely be stopped at 13:
```python
def primes():
whlPrms = [2,3,5,7,11,13,17] # base wheel primes
for p in whlPrms: yield p
def makeGaps():
buf = [True] * (3 * 5 * 7 * 11 * 13 * 17 + 1) # all odds plus extra for o/f
for p in whlPrms:
if p < 3:
continue # no need to handle evens
strt = (p * p - 19) >> 1 # start position (divided by 2 using shift)
while strt < 0: strt += p
buf[strt::p] = [False] * ((len(buf) - strt - 1) // p + 1) # cull for p
whlPsns = [i + i for i,v in enumerate(buf) if v]
return [whlPsns[i + 1] - whlPsns[i] for i in range(len(whlPsns) - 1)]
gaps = makeGaps() # big wheel gaps
def wheel_prime_pairs():
yield (19,0); bps = wheel_prime_pairs() # additional primes supply
p, pi = next(bps); q = p * p # adv to get 11 sqr'd is 121 as next square to put
sieve = {}; n = 23; ni = 1 # into sieve dict; init cndidate, wheel ndx
while True:
if n not in sieve: # is not a multiple of previously recorded primes
if n < q: yield (n, ni) # n is prime with wheel modulo index
else:
npi = pi + 1 # advance wheel index
if npi > 92159: npi = 0
sieve[q + p * gaps[pi]] = (p, npi) # n == p * p: put next cull position on wheel
p, pi = next(bps); q = p * p # advance next prime and prime square to put
else:
s, si = sieve.pop(n)
nxt = n + s * gaps[si] # move current cull position up the wheel
si = si + 1 # advance wheel index
if si > 92159: si = 0
while nxt in sieve: # ensure each entry is unique by wheel
nxt += s * gaps[si]
si = si + 1 # advance wheel index
if si > 92159: si = 0
sieve[nxt] = (s, si) # next non-marked multiple of a prime
nni = ni + 1 # advance wheel index
if nni > 92159: nni = 0
n += gaps[ni]; ni = nni # advance on the wheel
for p, pi in wheel_prime_pairs(): yield p # strip out indexes
```
### Iterative sieve on unbounded count from 2
See [[Extensible_prime_generator#Python:_Iterative_sieve_on_unbounded_count_from_2| Extensible prime generator: Iterative sieve on unbounded count from 2]]
## R
```r
sieve <- function(n) {
if (n < 2) return(NULL)
a <- rep(T, n)
a[1] <- F
for(i in seq(n)) {
if (a[i]) {
j <- i * i
if (j > n) return(which(a))
a[seq(j, n, by=i)] <- F
}
}
}
sieve(1000)
```
## Racket
### Imperative versions
Ugly imperative version:
```Racket
#lang racket
(define (sieve n)
(define non-primes '())
(define primes '())
(for ([i (in-range 2 (add1 n))])
(unless (member i non-primes)
(set! primes (cons i primes))
(for ([j (in-range (* i i) (add1 n) i)])
(set! non-primes (cons j non-primes)))))
(reverse primes))
(sieve 100)
```
A little nicer, but still imperative:
```Racket
#lang racket
(define (sieve n)
(define primes (make-vector (add1 n) #t))
(for* ([i (in-range 2 (add1 n))]
#:when (vector-ref primes i)
[j (in-range (* i i) (add1 n) i)])
(vector-set! primes j #f))
(for/list ([n (in-range 2 (add1 n))]
#:when (vector-ref primes n))
n))
(sieve 100)
```
Imperative version using a bit vector:
```Racket
#lang racket
(require data/bit-vector)
;; Returns a list of prime numbers up to natural number limit
(define (eratosthenes limit)
(define bv (make-bit-vector (+ limit 1) #f))
(bit-vector-set! bv 0 #t)
(bit-vector-set! bv 1 #t)
(for* ([i (in-range (add1 (sqrt limit)))] #:unless (bit-vector-ref bv i)
[j (in-range (* 2 i) (bit-vector-length bv) i)])
(bit-vector-set! bv j #t))
;; translate to a list of primes
(for/list ([i (bit-vector-length bv)] #:unless (bit-vector-ref bv i)) i))
(eratosthenes 100)
```
'(2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97)
### Infinite list of primes Using laziness
These examples use infinite lists (streams) to implement the sieve of Eratosthenes in a functional way, and producing all prime numbers. The following functions are used as a prefix for pieces of code that follow:
```Racket
#lang lazy
(define (ints-from i d) (cons i (ints-from (+ i d) d)))
(define (after n l f)
(if (< (car l) n) (cons (car l) (after n (cdr l) f)) (f l)))
(define (diff l1 l2)
(let ([x1 (car l1)] [x2 (car l2)])
(cond [(< x1 x2) (cons x1 (diff (cdr l1) l2 ))]
[(> x1 x2) (diff l1 (cdr l2)) ]
[else (diff (cdr l1) (cdr l2)) ])))
(define (union l1 l2) ; union of two lists
(let ([x1 (car l1)] [x2 (car l2)])
(cond [(< x1 x2) (cons x1 (union (cdr l1) l2 ))]
[(> x1 x2) (cons x2 (union l1 (cdr l2)))]
[else (cons x1 (union (cdr l1) (cdr l2)))])))
```
### = Basic sieve =
```Racket
(define (sieve l)
(define x (car l))
(cons x (sieve (diff (cdr l) (ints-from (+ x x) x)))))
(define primes (sieve (ints-from 2 1)))
(!! (take 25 primes))
```
Runs at ~ n^2.1 [http://en.wikipedia.org/wiki/Analysis_of_algorithms#Empirical_orders_of_growth empirically], for ''n <= 1500'' primes produced.
### = With merged composites =
Note that the first number, 2, and its multiples stream (ints-from 4 2) are handled separately to ensure that the non-primes list is never empty, which simplifies the code for union which assumes non-empty infinite lists.
```Racket
(define (sieve l non-primes)
(let ([x (car l)] [np (car non-primes)])
(cond [(= x np) (sieve (cdr l) (cdr non-primes))] ; else x < np
[else (cons x (sieve (cdr l) (union (ints-from (* x x) x)
non-primes)))])))
(define primes (cons 2 (sieve (ints-from 3 1) (ints-from 4 2))))
```
### = Basic sieve Optimized with postponed processing =
Since a prime's multiples that count start from its square, we should only start removing them when we reach that square.
```Racket
(define (sieve l prs)
(define p (car prs))
(define q (* p p))
(after q l (λ(t) (sieve (diff t (ints-from q p)) (cdr prs)))))
(define primes (cons 2 (sieve (ints-from 3 1) primes)))
```
Runs at ~ n^1.4 up to n=10,000. The initial 2 in the self-referential primes definition is needed to prevent a "black hole".
### = Merged composites Optimized with postponed processing =
Since prime's multiples that matter start from its square, we should only add them when we reach that square.
```Racket
(define (composites l q primes)
(after q l
(λ(t)
(let ([p (car primes)] [r (cadr primes)])
(composites (union t (ints-from q p)) ; q = p*p
(* r r) (cdr primes))))))
(define primes (cons 2
(diff (ints-from 3 1)
(composites (ints-from 4 2) 9 (cdr primes)))))
```
==== Implementation of Richard Bird's algorithm ====
Appears in [http://www.cs.hmc.edu/~oneill/papers/Sieve-JFP.pdf M.O'Neill's paper]. Achieves on its own the proper postponement that is specifically arranged for in the version above (with after), and is yet more efficient, because it folds to the right and so builds the right-leaning structure of merges at run time, where the more frequently-producing streams of multiples appear higher in that structure, so the composite numbers produced by them have less merge nodes to percolate through:
```Racket
(define primes
(cons 2 (diff (ints-from 3 1)
(foldr (λ(p r) (define q (* p p))
(cons q (union (ints-from (+ q p) p) r)))
'() primes))))
```
### Using threads and channels
Same algorithm as [[#With merged composites|"merged composites" above]] (without the postponement optimization), but now using threads and channels to produce a channel of all prime numbers (similar to newsqueak). The macro at the top is a convenient wrapper around definitions of channels using a thread that feeds them.
```Racket
#lang racket
(define-syntax (define-thread-loop stx)
(syntax-case stx ()
[(_ (name . args) expr ...)
(with-syntax ([out! (datum->syntax stx 'out!)])
#'(define (name . args)
(define out (make-channel))
(define (out! x) (channel-put out x))
(thread (λ() (let loop () expr ... (loop))))
out))]))
(define-thread-loop (ints-from i d) (out! i) (set! i (+ i d)))
(define-thread-loop (merge c1 c2)
(let loop ([x1 (channel-get c1)] [x2 (channel-get c2)])
(cond [(> x1 x2) (out! x2) (loop x1 (channel-get c2))]
[(< x1 x2) (out! x1) (loop (channel-get c1) x2)]
[else (out! x1) (loop (channel-get c1) (channel-get c2))])))
(define-thread-loop (sieve l non-primes)
(let loop ([x (channel-get l)] [np (channel-get non-primes)])
(cond [(> x np) (loop x (channel-get non-primes))]
[(= x np) (loop (channel-get l) (channel-get non-primes))]
[else (out! x)
(set! non-primes (merge (ints-from (* x x) x) non-primes))
(loop (channel-get l) np)])))
(define-thread-loop (cons x l)
(out! x) (let loop () (out! (channel-get l)) (loop)))
(define primes (cons 2 (sieve (ints-from 3 1) (ints-from 4 2))))
(for/list ([i 25] [x (in-producer channel-get eof primes)]) x)
```
### Using generators
Yet another variation of the same algorithm as above, this time using generators.
```Racket
#lang racket
(require racket/generator)
(define (ints-from i d)
(generator () (let loop ([i i]) (yield i) (loop (+ i d)))))
(define (merge g1 g2)
(generator ()
(let loop ([x1 (g1)] [x2 (g2)])
(cond [(< x1 x2) (yield x1) (loop (g1) x2)]
[(> x1 x2) (yield x2) (loop x1 (g2))]
[else (yield x1) (loop (g1) (g2))]))))
(define (sieve l non-primes)
(generator ()
(let loop ([x (l)] [np (non-primes)])
(cond [(> x np) (loop x (non-primes))]
[(= x np) (loop (l) (non-primes))]
[else (yield x)
(set! non-primes (merge (ints-from (* x x) x) non-primes))
(loop (l) np)]))))
(define (cons x l) (generator () (yield x) (let loop () (yield (l)) (loop))))
(define primes (cons 2 (sieve (ints-from 3 1) (ints-from 4 2))))
(for/list ([i 25] [x (in-producer primes)]) x)
```
## REXX
### no wheel version
The first three REXX versions make use of a sparse stemmed array: ['''@.'''].
As the stemmed array gets heavily populated, the number of entries ''may'' slow down the REXX interpreter substantially,
depending upon the efficacy of the hashing technique being used for REXX variables (setting/retrieving).
```REXX
/*REXX program generates and displays primes via the sieve of Eratosthenes algorithm.*/
parse arg H .; if H=='' | H=="," then H= 200 /*optain optional argument from the CL.*/
w= length(H); @prime= right('prime', 20) /*W: is used for aligning the output.*/
@.=. /*assume all the numbers are prime. */
#= 0 /*number of primes found (so far). */
do j=2 for H-1; if @.j=='' then iterate /*all prime integers up to H inclusive.*/
#= # + 1 /*bump the prime number counter. */
say @prime right(#,w) " ───► " right(j,w) /*display the prime to the terminal. */
do m=j*j to H by j; @.m=; end /*strike all multiples as being ¬ prime*/
end /*j*/ /* ─── */
say /*stick a fork in it, we're all done. */
say right(#, 1+w+length(@prime) ) 'primes found up to and including ' H
```
'''output''' when using the input default of: 200
prime 1 ───► 2
prime 2 ───► 3
prime 3 ───► 5
prime 4 ───► 7
prime 5 ───► 11
prime 6 ───► 13
prime 7 ───► 17
prime 8 ───► 19
prime 9 ───► 23
prime 10 ───► 29
prime 11 ───► 31
prime 12 ───► 37
prime 13 ───► 41
prime 14 ───► 43
prime 15 ───► 47
prime 16 ───► 53
prime 17 ───► 59
prime 18 ───► 61
prime 19 ───► 67
prime 20 ───► 71
prime 21 ───► 73
prime 22 ───► 79
prime 23 ───► 83
prime 24 ───► 89
prime 25 ───► 97
prime 26 ───► 101
prime 27 ───► 103
prime 28 ───► 107
prime 29 ───► 109
prime 30 ───► 113
prime 31 ───► 127
prime 32 ───► 131
prime 33 ───► 137
prime 34 ───► 139
prime 35 ───► 149
prime 36 ───► 151
prime 37 ───► 157
prime 38 ───► 163
prime 39 ───► 167
prime 40 ───► 173
prime 41 ───► 179
prime 42 ───► 181
prime 43 ───► 191
prime 44 ───► 193
prime 45 ───► 197
prime 46 ───► 199
46 primes found up to and including 200
```
===wheel version, optional prime list suppression===
This version skips striking the even numbers (as being not prime), '''2''' is handled as a special case.
Also supported is the suppression of listing the primes if the '''H''' ('''h'''igh limit) is negative.
Also added is a final message indicating the number of primes found.
```rexx
/*REXX program generates primes via a wheeled sieve of Eratosthenes algorithm. */
parse arg H .; if H=='' then H=200 /*let the highest number be specified. */
tell=h>0; H=abs(H); w=length(H) /*a negative H suppresses prime listing*/
if 2<=H & tell then say right(1, w+20)'st prime ───► ' right(2, w)
@.= '0'x /*assume that all numbers are prime. */
cw= length(@.) /*the cell width that holds numbers. */
#= w<=H /*the number of primes found (so far).*/
!=0 /*skips the top part of sieve marking. */
do j=3 by 2 for (H-2)%2; b= j%cw /*odd integers up to H inclusive. */
if substr(x2b(c2x(@.b)),j//cw+1,1) then iterate /*is J composite ? */
#= # + 1 /*bump the prime number counter. */
if tell then say right(#, w+20)th(#) 'prime ───► ' right(j, w)
if ! then iterate /*should the top part be skipped ? */
jj=j * j /*compute the square of J. ___*/
if jj>H then !=1 /*indicates skip top part if j > √ H */
do m=jj to H by j+j; call . m; end /* [↑] strike odd multiples ¬ prime */
end /*j*/ /* ─── */
say; say right(#, w+20) 'prime's(#) "found up to and including " H
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────────────*/
.: parse arg n; b=n%cw; r=n//cw+1;_=x2b(c2x(@.b));@.b=x2c(b2x(left(_,r-1)'1'substr(_,r+1)));return
s: if arg(1)==1 then return arg(3); return word(arg(2) 's',1) /*pluralizer.*/
th: procedure; parse arg x; x=abs(x); return word('th st nd rd',1+x//10*(x//100%10\==1)*(x//10<4))
```
1st prime ───► 2
2nd prime ───► 3
3rd prime ───► 5
4th prime ───► 7
5th prime ───► 11
6th prime ───► 13
7th prime ───► 17
8th prime ───► 19
9th prime ───► 23
10th prime ───► 29
11th prime ───► 31
12th prime ───► 37
13th prime ───► 41
14th prime ───► 43
15th prime ───► 47
16th prime ───► 53
17th prime ───► 59
18th prime ───► 61
19th prime ───► 67
20th prime ───► 71
21st prime ───► 73
22nd prime ───► 79
23rd prime ───► 83
24th prime ───► 89
25th prime ───► 97
26th prime ───► 101
27th prime ───► 103
28th prime ───► 107
29th prime ───► 109
30th prime ───► 113
31st prime ───► 127
32nd prime ───► 131
33rd prime ───► 137
34th prime ───► 139
35th prime ───► 149
36th prime ───► 151
37th prime ───► 157
38th prime ───► 163
39th prime ───► 167
40th prime ───► 173
41st prime ───► 179
42nd prime ───► 181
43rd prime ───► 191
44th prime ───► 193
45th prime ───► 197
46th prime ───► 199
46 primes found up to and including 200
```
```txt
168 primes found up to and including 1000
```
```txt
1229 primes found up to and including 10000
```
9592 primes found up to and including 100000.
```
```txt
78498 primes found up to and including 10000000
```
```txt
664579 primes found up to and including 10000000
```
### wheel version
This version skips striking the even numbers (as being not prime), '''2''' is handled as a special case.
It also uses a short-circuit test for striking out composites ≤ √{{overline| target }}
```rexx
/*REXX pgm generates and displays primes via a wheeled sieve of Eratosthenes algorithm. */
parse arg H .; if H=='' | H=="," then H= 200 /*obtain the optional argument from CL.*/
w= length(H); @prime= right('prime', 20) /*w: is used for aligning the output. */
if 2<=H then say @prime right(1, w) " ───► " right(2, w)
#= 2<=H /*the number of primes found (so far).*/
@.=. /*assume all the numbers are prime. */
!=0; do j=3 by 2 for (H-2)%2 /*the odd integers up to H inclusive.*/
if @.j=='' then iterate /*Is composite? Then skip this number.*/
#= # + 1 /*bump the prime number counter. */
say @prime right(#,w) " ───► " right(j,w) /*display the prime to the terminal. */
if ! then iterate /*skip the top part of loop? ___ */
if j*j>H then !=1 /*indicate skip top part if J > √ H */
do m=j*j to H by j+j; @.m=; end /*strike odd multiples as not prime. */
end /*j*/ /* ─── */
say /*stick a fork in it, we're all done. */
say right(#, 1 + w + length(@prime) ) 'primes found up to and including ' H
```
The addition of the short-circuit test (using the REXX variable '''!''') makes it about ''another'' '''20%''' faster.
### Wheel Version restructured
```rexx
/*REXX program generates primes via sieve of Eratosthenes algorithm.
* 21.07.2012 Walter Pachl derived from above Rexx version
* avoid symbols @ and # (not supported by ooRexx)
* avoid negations (think positive)
**********************************************************************/
highest=200 /*define highest number to use. */
is_prime.=1 /*assume all numbers are prime. */
w=length(highest) /*width of the biggest number, */
/* it's used for aligned output.*/
Do j=3 To highest By 2, /*strike multiples of odd ints. */
While j*j<=highest /* up to sqrt(highest) */
If is_prime.j Then Do
Do jm=j*3 To highest By j+j /*start with next odd mult. of J */
is_prime.jm=0 /*mark odd mult. of J not prime. */
End
End
End
np=0 /*number of primes shown */
Call tell 2
Do n=3 To highest By 2 /*list all the primes found. */
If is_prime.n Then Do
Call tell n
End
End
Exit
tell: Parse Arg prime
np=np+1
Say ' prime number' right(np,w) " --> " right(prime,w)
Return
```
'''output''' is mostly identical to the above versions.
## Ring
```ring
limit = 100
sieve = list(limit)
for i = 2 to limit
for k = i*i to limit step i
sieve[k] = 1
next
if sieve[i] = 0 see "" + i + " " ok
next
```
Output:
```txt
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
```
## Ruby
''eratosthenes'' starts with nums = [nil, nil, 2, 3, 4, 5, ..., n], then marks ( the nil setting ) multiples of 2, 3, 5, 7, ... there, then returns all non-nil numbers which are the primes.
```ruby
def eratosthenes(n)
nums = [nil, nil, *2..n]
(2..Math.sqrt(n)).each do |i|
(i**2..n).step(i){|m| nums[m] = nil} if nums[i]
end
nums.compact
end
p eratosthenes(100)
```
```txt
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]
```
### With a wheel
''eratosthenes2'' adds more optimizations, but the code is longer.
* The array nums only tracks odd numbers (skips multiples of 2).
* The array nums holds booleans instead of integers, and every multiple of 3 begins false.
* The outer loop skips multiples of 2 and 3.
* Both inner loops skip multiples of 2 and 3.
```ruby
def eratosthenes2(n)
# For odd i, if i is prime, nums[i >> 1] is true.
# Set false for all multiples of 3.
nums = [true, false, true] * ((n + 5) / 6)
nums[0] = false # 1 is not prime.
nums[1] = true # 3 is prime.
# Outer loop and both inner loops are skipping multiples of 2 and 3.
# Outer loop checks i * i > n, same as i > Math.sqrt(n).
i = 5
until (m = i * i) > n
if nums[i >> 1]
i_times_2 = i << 1
i_times_4 = i << 2
while m <= n
nums[m >> 1] = false
m += i_times_2
nums[m >> 1] = false
m += i_times_4 # When i = 5, skip 45, 75, 105, ...
end
end
i += 2
if nums[i >> 1]
m = i * i
i_times_2 = i << 1
i_times_4 = i << 2
while m <= n
nums[m >> 1] = false
m += i_times_4 # When i = 7, skip 63, 105, 147, ...
nums[m >> 1] = false
m += i_times_2
end
end
i += 4
end
primes = [2]
nums.each_index {|i| primes << (i * 2 + 1) if nums[i]}
primes.pop while primes.last > n
primes
end
p eratosthenes2(100)
```
This simple benchmark compares ''eratosthenes'' with ''eratosthenes2''.
```ruby
require 'benchmark'
Benchmark.bmbm {|x|
x.report("eratosthenes") { eratosthenes(1_000_000) }
x.report("eratosthenes2") { eratosthenes2(1_000_000) }
}
```
''eratosthenes2'' runs about 4 times faster than ''eratosthenes''.
### With the standard library
[[MRI]] 1.9.x implements the sieve of Eratosthenes at file [http://redmine.ruby-lang.org/projects/ruby-19/repository/entry/lib/prime.rb prime.rb], class EratosthensesSeive (around [http://redmine.ruby-lang.org/projects/ruby-19/repository/entry/lib/prime.rb#L421 line 421]). This implementation optimizes for space, by packing the booleans into 16-bit integers. It also hardcodes all primes less than 256.
```ruby
require 'prime'
p Prime::EratosthenesGenerator.new.take_while {|i| i <= 100}
```
## Run BASIC
```runbasic
input "Gimme the limit:"; limit
dim flags(limit)
for i = 2 to limit
for k = i*i to limit step i
flags(k) = 1
next k
if flags(i) = 0 then print i;", ";
next i
```
```txt
Gimme the limit:?100
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97,
```
## Rust
===Sieve of Eratosthenes - No optimization===
```rust
fn simple_sieve(limit: usize) -> Vec {
let mut is_prime = vec![true; limit+1];
is_prime[0] = false;
if limit >= 1 { is_prime[1] = false }
for num in 2..limit+1 {
if is_prime[num] {
let mut multiple = num*num;
while multiple <= limit {
is_prime[multiple] = false;
multiple += num;
}
}
}
is_prime.iter().enumerate()
.filter_map(|(pr, &is_pr)| if is_pr {Some(pr)} else {None} )
.collect()
}
fn main() {
println!("{:?}", simple_sieve(100));
}
```
```txt
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]
```
===Basic Version slightly optimized, Iterator output===
The above code doesn't even do the basic optimizing of only culling composites by primes up to the square root of the range as allowed in the task; it also outputs a vector of resulting primes, which consumes memory. The following code fixes both of those, outputting the results as an Iterator:
```rust
use std::iter::{empty, once};
use std::time::Instant;
fn basic_sieve(limit: usize) -> Box> {
if limit < 2 { return Box::new(empty()) }
let mut is_prime = vec![true; limit+1];
is_prime[0] = false;
if limit >= 1 { is_prime[1] = false }
let sqrtlmt = (limit as f64).sqrt() as usize + 1;
for num in 2..sqrtlmt {
if is_prime[num] {
let mut multiple = num * num;
while multiple <= limit {
is_prime[multiple] = false;
multiple += num;
}
}
}
Box::new(is_prime.into_iter().enumerate()
.filter_map(|(p, is_prm)| if is_prm { Some(p) } else { None }))
}
fn main() {
let n = 1000000;
let vrslt = basic_sieve(100).collect::>();
println!("{:?}", vrslt);
let strt = Instant::now();
// do it 1000 times to get a reasonable execution time span...
let rslt = (1..1000).map(|_| basic_sieve(n)).last().unwrap();
let elpsd = strt.elapsed();
let count = rslt.count();
println!("{}", count);
let secs = elpsd.as_secs();
let millis = (elpsd.subsec_nanos() / 1000000) as u64;
let dur = secs * 1000 + millis;
println!("Culling composites took {} milliseconds.", dur);
}
```
```txt
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]
78498
Culling composites took 4595 milliseconds.
```
The sieving operation is run for 1000 loops in order to get a reasonable execution time for comparison.
===Odds-only bit-packed array, Iterator output===
The following code improves the above code by sieving only odd composite numbers as 2 is the only even prime for a reduction in number of operations by a factor of about two and a half with reduction of memory use by a factor of two, and bit-packs the composite sieving array for a further reduction of memory use by a factor of eight and with some saving in time due to better CPU cache use for a given sieving range; it also demonstrates how to eliminate the redundant array bounds check:
```rust
fn optimized_sieve(limit: usize) -> Box> {
if limit < 3 {
return if limit < 2 { Box::new(empty()) } else { Box::new(once(2)) }
}
let ndxlmt = (limit - 3) / 2 + 1;
let bfsz = ((limit - 3) / 2) / 32 + 1;
let mut cmpsts = vec![0u32; bfsz];
let sqrtndxlmt = ((limit as f64).sqrt() as usize - 3) / 2 + 1;
for ndx in 0..sqrtndxlmt {
if (cmpsts[ndx >> 5] & (1u32 << (ndx & 31))) == 0 {
let p = ndx + ndx + 3;
let mut cullpos = (p * p - 3) / 2;
while cullpos < ndxlmt {
unsafe { // avoids array bounds check, which is already done above
let cptr = cmpsts.get_unchecked_mut(cullpos >> 5);
*cptr |= 1u32 << (cullpos & 31);
}
// cmpsts[cullpos >> 5] |= 1u32 << (cullpos & 31); // with bounds check
cullpos += p;
}
}
}
Box::new((-1 .. ndxlmt as isize).into_iter().filter_map(move |i| {
if i < 0 { Some(2) } else {
if cmpsts[i as usize >> 5] & (1u32 << (i & 31)) == 0 {
Some((i + i + 3) as usize) } else { None } }
}))
}
```
The above function can be used just by substituting "optimized_sieve" for "basic_sieve" in the previous "main" function, and the outputs are the same except that the time is only 1584 milliseconds, or about three times as fast.
===Unbounded Page-Segmented bit-packed odds-only version with Iterator===
'''Caution!''' This implementation is used in the [[Extensible_prime_generator#Rust|Extensible prime generator task]], so be sure not to break that implementation when changing this code.
While that above code is quite fast, as the range increases above the 10's of millions it begins to lose efficiency due to loss of CPU cache associativity as the size of the one-large-array used for culling composites grows beyond the limits of the various CPU caches. Accordingly the following page-segmented code where each culling page can be limited to not larger than the L1 CPU cache is about four times faster than the above for the range of one billion:
```rust
use std::iter::{empty, once};
use std::rc::Rc;
use std::cell::RefCell;
use std::time::Instant;
const RANGE: u64 = 1000000000;
const SZ_PAGE_BTS: u64 = (1 << 14) * 8; // this should be the size of the CPU L1 cache
const SZ_BASE_BTS: u64 = (1 << 7) * 8;
static CLUT: [u8; 256] = [
8, 7, 7, 6, 7, 6, 6, 5, 7, 6, 6, 5, 6, 5, 5, 4, 7, 6, 6, 5, 6, 5, 5, 4, 6, 5, 5, 4, 5, 4, 4, 3,
7, 6, 6, 5, 6, 5, 5, 4, 6, 5, 5, 4, 5, 4, 4, 3, 6, 5, 5, 4, 5, 4, 4, 3, 5, 4, 4, 3, 4, 3, 3, 2,
7, 6, 6, 5, 6, 5, 5, 4, 6, 5, 5, 4, 5, 4, 4, 3, 6, 5, 5, 4, 5, 4, 4, 3, 5, 4, 4, 3, 4, 3, 3, 2,
6, 5, 5, 4, 5, 4, 4, 3, 5, 4, 4, 3, 4, 3, 3, 2, 5, 4, 4, 3, 4, 3, 3, 2, 4, 3, 3, 2, 3, 2, 2, 1,
7, 6, 6, 5, 6, 5, 5, 4, 6, 5, 5, 4, 5, 4, 4, 3, 6, 5, 5, 4, 5, 4, 4, 3, 5, 4, 4, 3, 4, 3, 3, 2,
6, 5, 5, 4, 5, 4, 4, 3, 5, 4, 4, 3, 4, 3, 3, 2, 5, 4, 4, 3, 4, 3, 3, 2, 4, 3, 3, 2, 3, 2, 2, 1,
6, 5, 5, 4, 5, 4, 4, 3, 5, 4, 4, 3, 4, 3, 3, 2, 5, 4, 4, 3, 4, 3, 3, 2, 4, 3, 3, 2, 3, 2, 2, 1,
5, 4, 4, 3, 4, 3, 3, 2, 4, 3, 3, 2, 3, 2, 2, 1, 4, 3, 3, 2, 3, 2, 2, 1, 3, 2, 2, 1, 2, 1, 1, 0 ];
fn count_page(lmti: usize, pg: &[u32]) -> i64 {
let pgsz = pg.len(); let pgbts = pgsz * 32;
let (lmt, icnt) = if lmti >= pgbts { (pgsz, 0) } else {
let lstw = lmti / 32;
let msk = 0xFFFFFFFEu32 << (lmti & 31);
let v = (msk | pg[lstw]) as usize;
(lstw, (CLUT[v & 0xFF] + CLUT[(v >> 8) & 0xFF]
+ CLUT[(v >> 16) & 0xFF] + CLUT[v >> 24]) as u32)
};
let mut count = 0u32;
for i in 0 .. lmt {
let v = pg[i] as usize;
count += (CLUT[v & 0xFF] + CLUT[(v >> 8) & 0xFF]
+ CLUT[(v >> 16) & 0xFF] + CLUT[v >> 24]) as u32;
}
(icnt + count) as i64
}
fn primes_pages() -> Box)>> {
// a memoized iterable enclosing a Vec that grows as needed from an Iterator...
type Bpasi = Box)>>; // (lwi, base cmpsts page)
type Bpas = Rc<(RefCell, RefCell>>)>; // interior mutables
struct Bps(Bpas); // iterable wrapper for base primes array state
struct Bpsi<'a>(usize, &'a Bpas); // iterator with current pos, state ref's
impl<'a> Iterator for Bpsi<'a> {
type Item = &'a Vec;
fn next(&mut self) -> Option {
let n = self.0; let bpas = self.1;
while n >= bpas.1.borrow().len() { // not thread safe
let nbpg = match bpas.0.borrow_mut().next() {
Some(v) => v, _ => (0, vec!()) };
if nbpg.1.is_empty() { return None } // end if no source iter
bpas.1.borrow_mut().push(cnvrt2bppg(nbpg));
}
self.0 += 1; // unsafe pointer extends interior -> exterior lifetime
// multi-threading might drop following Vec while reading - protect
let ptr = &bpas.1.borrow()[n] as *const Vec;
unsafe { Some(&(*ptr)) }
}
}
impl<'a> IntoIterator for &'a Bps {
type Item = &'a Vec;
type IntoIter = Bpsi<'a>;
fn into_iter(self) -> Self::IntoIter {
Bpsi(0, &self.0)
}
}
fn make_page(lwi: u64, szbts: u64, bppgs: &Bpas)
-> (u64, Vec) {
let nxti = lwi + szbts;
let pbts = szbts as usize;
let mut cmpsts = vec!(0u32; pbts / 32);
'outer: for bpg in Bps(bppgs.clone()).into_iter() { // in the inner tight loop...
let pgsz = bpg.len();
for i in 0 .. pgsz {
let p = bpg[i] as u64; let pc = p as usize;
let s = (p * p - 3) / 2;
if s >= nxti { break 'outer; } else { // page start address:
let mut cp = if s >= lwi { (s - lwi) as usize } else {
let r = ((lwi - s) % p) as usize;
if r == 0 { 0 } else { pc - r }
};
while cp < pbts {
unsafe { // avoids array bounds check, which is already done above
let cptr = cmpsts.get_unchecked_mut(cp >> 5);
*cptr |= 1u32 << (cp & 31); // about as fast as it gets...
}
// cmpsts[cp >> 5] |= 1u32 << (cp & 31);
cp += pc;
}
}
}
}
(lwi, cmpsts)
}
fn pages_from(lwi: u64, szbts: u64, bpas: Bpas)
-> Box)>> {
struct Gen(u64, u64);
impl Iterator for Gen {
type Item = (u64, u64);
#[inline]
fn next(&mut self) -> Option<(u64, u64)> {
let v = self.0; let inc = self.1; // calculate variable size here
self.0 = v + inc;
Some((v, inc))
}
}
Box::new(Gen(lwi, szbts)
.map(move |(lwi, szbts)| make_page(lwi, szbts, &bpas)))
}
fn cnvrt2bppg(cmpsts: (u64, Vec)) -> Vec {
let (lwi, pg) = cmpsts;
let pgbts = pg.len() * 32;
let cnt = count_page(pgbts, &pg) as usize;
let mut bpv = vec!(0u32; cnt);
let mut j = 0; let bsp = (lwi + lwi + 3) as usize;
for i in 0 .. pgbts {
if (pg[i >> 5] & (1u32 << (i & 0x1F))) == 0u32 {
bpv[j] = (bsp + i + i) as u32; j += 1;
}
}
bpv
}
// recursive Rc/RefCell variable bpas - used only for init, then fixed ...
// start with just enough base primes to init the first base primes page...
let base_base_prms = vec!(3u32,5u32,7u32);
let rcvv = RefCell::new(vec!(base_base_prms));
let bpas: Bpas = Rc::new((RefCell::new(Box::new(empty())), rcvv));
let initpg = make_page(0, 32, &bpas); // small base primes page for SZ_BASE_BTS = 2^7 * 8
*bpas.1.borrow_mut() = vec!(cnvrt2bppg(initpg)); // use for first page
let frstpg = make_page(0, SZ_BASE_BTS, &bpas); // init bpas for first base prime page
*bpas.0.borrow_mut() = pages_from(SZ_BASE_BTS, SZ_BASE_BTS, bpas.clone()); // recurse bpas
*bpas.1.borrow_mut() = vec!(cnvrt2bppg(frstpg)); // fixed for subsequent pages
pages_from(0, SZ_PAGE_BTS, bpas) // and bpas also used here for main pages
}
fn primes_paged() -> Box