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Two integers $N$ and $M$ are said to be [[wp:Amicable numbers|amicable pairs]] if $N \neq M$ and the sum of the [[Proper divisors|proper divisors]] of $N$ ($\mathrm\left\{sum\right\}\left(\mathrm\left\{propDivs\right\}\left(N\right)\right)$) $= M$ as well as $\mathrm\left\{sum\right\}\left(\mathrm\left\{propDivs\right\}\left(M\right)\right) = N$.

;Example: '''1184''' and '''1210''' are an amicable pair, with proper divisors:

• 1, 2, 4, 8, 16, 32, 37, 74, 148, 296, 592 and
• 1, 2, 5, 10, 11, 22, 55, 110, 121, 242, 605 respectively.

;Task: Calculate and show here the Amicable pairs below 20,000; (there are eight).

• [[Proper divisors]]
• [[Abundant, deficient and perfect number classifications]]
• [[Aliquot sequence classifications]] and its amicable ''classification''.

## 11l

F sum_proper_divisors(n)
R I n < 2 {0} E sum((1 .. n I/ 2).filter(it -> (@n % it) == 0))

L(n) 1..20000
V m = sum_proper_divisors(n)
I m > n & sum_proper_divisors(m) == n
print(n"\t"m)


This solution uses the package ''Generic_Divisors'' from the Proper Divisors task [[http://rosettacode.org/wiki/Proper_divisors#Ada]].

with Ada.Text_IO, Generic_Divisors; use Ada.Text_IO;

procedure Amicable_Pairs is

function Same(P: Positive) return Positive is (P);

package Divisor_Sum is new Generic_Divisors
(Result_Type => Natural, None => 0, One => Same, Add =>  "+");

Num2 : Integer;
begin
for Num1 in 4 .. 20_000 loop
Num2 := Divisor_Sum.Process(Num1);
if Num1 < Num2 then
if Num1 = Divisor_Sum.Process(Num2) then
Put_Line(Integer'Image(Num1) & "," & Integer'Image(Num2));
end if;
end if;
end loop;
end Amicable_Pairs;


{{Out}}


220, 284
1184, 1210
2620, 2924
5020, 5564
6232, 6368
10744, 10856
12285, 14595
17296, 18416



## ALGOL 68

# resturns the sum of the proper divisors of n                    #
# if n = 1, 0 or -1, we return 0                                  #
PROC sum proper divisors = ( INT n )INT:
BEGIN
INT result := 0;
INT abs n = ABS n;
IF abs n > 1 THEN
FOR d FROM ENTIER sqrt( abs n ) BY -1 TO 2 DO
IF abs n MOD d = 0 THEN
# found another divisor                      #
result +:= d;
IF d * d /= n THEN
# include the other divisor              #
result +:= n OVER d
FI
FI
OD;
# 1 is always a proper divisor of numbers > 1        #
result +:= 1
FI;
result
END # sum proper divisors # ;

# construct a table of the sum of the proper divisors of numbers  #
# up to 20 000                                                    #
INT max number = 20 000;
[ 1 : max number ]INT proper divisor sum;
FOR n TO UPB proper divisor sum DO proper divisor sum[ n ] := sum proper divisors( n ) OD;

# returns TRUE if n1 and n2 are an amicable pair FALSE otherwise  #
#         n1 and n2 are amicable if the sum of the proper diviors #
#         n1 = n2 and the sum of the proper divisors of n2 = n1   #
PROC is an amicable pair = ( INT n1, n2 )BOOL:
( proper divisor sum[ n1 ] = n2 AND proper divisor sum[ n2 ] = n1 );

# find the amicable pairs up to 20 000                            #
FOR p1 TO max number DO
FOR p2 FROM p1 + 1 TO max number DO
IF is an amicable pair( p1, p2 ) THEN
print( ( whole( p1, -6 ), " and ", whole( p2, -6 ), " are a amicable pair", newline ) )
FI
OD
OD


{{out}}


220 and    284 are a amicable pair
1184 and   1210 are a amicable pair
2620 and   2924 are a amicable pair
5020 and   5564 are a amicable pair
6232 and   6368 are a amicable pair
10744 and  10856 are a amicable pair
12285 and  14595 are a amicable pair
17296 and  18416 are a amicable pair



## ANSI Standard BASIC

{{Trans|GFA Basic}}

100 DECLARE EXTERNAL FUNCTION sum_proper_divisors
110 CLEAR
120 !
130 DIM f(20001)      ! sum of proper factors for each n
140 FOR i=1 TO 20000
150    LET f(i)=sum_proper_divisors(i)
160 NEXT i
170 ! look for pairs
180 FOR i=1 TO 20000
190    FOR j=i+1 TO 20000
200       IF f(i)=j AND i=f(j) THEN
210          PRINT "Amicable pair ";i;" ";j
220       END IF
230    NEXT j
240 NEXT i
250 !
260 PRINT
270 PRINT "-- found all amicable pairs"
280 END
290 !
300 ! Compute the sum of proper divisors of given number
310 !
320 EXTERNAL FUNCTION sum_proper_divisors(n)
330 !
340 IF n>1 THEN ! n must be 2 or larger
360    LET root=SQR(n)    ! note that root is an integer
370    ! check possible factors, up to sqrt
380    FOR i=2 TO root
390       IF MOD(n,i)=0 THEN
400          LET sum=sum+i     ! i is a factor
410          IF i*i<>n THEN    ! check i is not actual square root of n
420             LET sum=sum+n/i  ! so n/i will also be a factor
430          END IF
440       END IF
450    NEXT i
460 END IF
470 LET sum_proper_divisors = sum
480 END FUNCTION


## AppleScript

{{Trans|JavaScript}}

-- AMICABLE PAIRS ------------------------------------------------------------

-- amicablePairsUpTo :: Int -> Int
on amicablePairsUpTo(max)

-- amicable :: [Int] -> Int -> Int -> [Int] -> [Int]
script amicable
on |λ|(a, m, n, lstSums)
if (m > n) and (m ≤ max) and ((item m of lstSums) = n) then
a & [[n, m]]
else
a
end if
end |λ|
end script

-- divisorsSummed :: Int -> Int
-- sum :: Int -> Int -> Int
script sum
on |λ|(a, b)
a + b
end |λ|
end script

on |λ|(n)
foldl(sum, 0, properDivisors(n))
end |λ|
end script

foldl(amicable, {}, ¬
end amicablePairsUpTo

-- TEST ----------------------------------------------------------------------
on run

amicablePairsUpTo(20000)

end run

-- PROPER DIVISORS -----------------------------------------------------------

-- properDivisors :: Int -> [Int]
on properDivisors(n)

-- isFactor :: Int -> Bool
script isFactor
on |λ|(x)
n mod x = 0
end |λ|
end script

-- integerQuotient :: Int -> Int
script integerQuotient
on |λ|(x)
(n / x) as integer
end |λ|
end script

if n = 1 then
{1}
else
set realRoot to n ^ (1 / 2)
set intRoot to realRoot as integer
set blnPerfectSquare to intRoot = realRoot

-- Factors up to square root of n,
set lows to filter(isFactor, enumFromTo(1, intRoot))

-- and quotients of these factors beyond the square root,
-- excluding n itself (last item)
items 1 thru -2 of (lows & map(integerQuotient, ¬
items (1 + (blnPerfectSquare as integer)) thru -1 of reverse of lows))
end if
end properDivisors

-- GENERIC FUNCTIONS ---------------------------------------------------------

-- enumFromTo :: Int -> Int -> [Int]
on enumFromTo(m, n)
if m > n then
set d to -1
else
set d to 1
end if
set lst to {}
repeat with i from m to n by d
set end of lst to i
end repeat
return lst
end enumFromTo

-- filter :: (a -> Bool) -> [a] -> [a]
on filter(f, xs)
tell mReturn(f)
set lst to {}
set lng to length of xs
repeat with i from 1 to lng
set v to item i of xs
if |λ|(v, i, xs) then set end of lst to v
end repeat
return lst
end tell
end filter

-- foldl :: (a -> b -> a) -> a -> [b] -> a
on foldl(f, startValue, xs)
tell mReturn(f)
set v to startValue
set lng to length of xs
repeat with i from 1 to lng
set v to |λ|(v, item i of xs, i, xs)
end repeat
return v
end tell
end foldl

-- map :: (a -> b) -> [a] -> [b]
on map(f, xs)
tell mReturn(f)
set lng to length of xs
set lst to {}
repeat with i from 1 to lng
set end of lst to |λ|(item i of xs, i, xs)
end repeat
return lst
end tell
end map

-- Lift 2nd class handler function into 1st class script wrapper
-- mReturn :: Handler -> Script
on mReturn(f)
if class of f is script then
f
else
script
property |λ| : f
end script
end if
end mReturn


{{Out}}

{{220, 284}, {1184, 1210}, {2620, 2924}, {5020, 5564},
{6232, 6368}, {10744, 10856}, {12285, 14595}, {17296, 18416}}


## ATS


(* ****** ****** *)
//
#include
#include
//
(* ****** ****** *)
//
fun
sum_list_vt
(xs: List_vt(int)): int =
(
case+ xs of
| ~list_vt_nil() => 0
| ~list_vt_cons(x, xs) => x + sum_list_vt(xs)
)
//
(* ****** ****** *)

fun
propDivs
(
x0: int
) : List0_vt(int) =
loop(x0, 2, list_vt_sing(1)) where
{
//
fun
loop
(
x0: int, i: int, res: List0_vt(int)
) : List0_vt(int) =
(
if
(i * i) > x0
then list_vt_reverse(res)
else
(
if x0 % i != 0
then
loop(x0, i+1, res)
// end of [then]
else let
val res =
cons_vt(i, res)
// end of [val]
val res =
(
if i * i = x0 then res else cons_vt(x0 / i, res)
) : List0_vt(int) // end of [val]
in
loop(x0, i+1, res)
end // end of [else]
// end of [if]
)
) (* end of [loop] *)
//
} // end of [propDivs]

(* ****** ****** *)

fun
sum_propDivs(x: int): int = sum_list_vt(propDivs(x))

(* ****** ****** *)

val
theNat2 = auxmain(2) where
{
fun
auxmain
(
n: int
) : stream_vt(int) = $ldelay(stream_vt_cons(n, auxmain(n+1))) } (* ****** ****** *) // val theAmicable = ( stream_vt_takeLte(theNat2, 20000) ).filter() ( lam x => let val x2 = sum_propDivs(x) in x < x2 && x = sum_propDivs(x2) end ) // (* ****** ****** *) val () = theAmicable.foreach() ( lam x => println! ("(", x, ", ", sum_propDivs(x), ")") ) (* ****** ****** *) implement main0 () = () (* ****** ****** *)  {{out}}  (220, 284) (1184, 1210) (2620, 2924) (5020, 5564) (6232, 6368) (10744, 10856) (12285, 14595) (17296, 18416)  ## AutoHotkey SetBatchLines -1 Loop, 20000 { m := A_index ; Getting factors loop % floor(sqrt(m)) { if ( mod(m, A_index) = 0 ) { if ( A_index ** 2 == m ) { sum += A_index continue } else if ( A_index != 1 ) { sum += A_index + m//A_index } else if ( A_index = 1 ) { sum += A_index } } } ; Factors obtained ; Checking factors of sum if ( sum > 1 ) { loop % floor(sqrt(sum)) { if ( mod(sum, A_index) = 0 ) { if ( A_index ** 2 == sum ) { sum2 += A_index continue } else if ( A_index != 1 ) { sum2 += A_index + sum//A_index } else if ( A_index = 1 ) { sum2 += A_index } } } if ( m = sum2 ) && ( m != sum ) && ( m < sum ) final .= m . ":" . sum . "n" } ; Checked sum := 0 sum2 := 0 } MsgBox % final ExitApp  {{out}}  220:284 1184:1210 2620:2924 5020:5564 6232:6368 10744:10856 12285:14595 17296:18416  ## AWK  #!/bin/awk -f function sumprop(num, i,sum,root) { if (num < 2) return 0 sum=1 root=sqrt(num) for ( i=2; i < root; i++) { if (num % i == 0 ) { sum = sum + i + num/i } } if (num % root == 0) { sum = sum + root } return sum } BEGIN{ limit=20000 print "Amicable pairs < ",limit for (n=1; n < limit+1; n++) { m=sumprop(n) if (n == sumprop(m) && n < m) print n,m } } }  {{out}}  # ./amicable Amicable pairs < 20000 220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416  ## Befunge v_@#-*8*:"2":$_:#!2#*8#g*#6:#0*#!:#-*#<v>*/.55+,
1>:28*:*:*%\28*:*:*/06p28*:*:*/\2v %%^:*:<>*v
+|!:-1g60/*:*:*82::+**:*:<<>:#**#8:#<*^>.28*^8 :
:v>>*:*%/\28*:*:*%+\v>8+#$^#_+#\:#0<:\1/*:*2#< 2v^:*82\/*:*:*82:::_v#!%%*:*:*82\/*:*:*82::<_^#< >>06p:28*:*:**1+01-\>1+::28*:*:*/\28*:*:*%:*\!^  {{out}} 220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416  ## C Remark: Look at Pascal Alternative [[http://rosettacode.org/wiki/Amicable_pairs#Alternative]].You are using the same principle, so here too both numbers of the pair must be < top. The program will overflow and error in all sorts of ways when given a commandline argument >= UINT_MAX/2 (generally 2^31) #include <stdio.h> #include <stdlib.h> typedef unsigned int uint; int main(int argc, char **argv) { uint top = atoi(argv[1]); uint *divsum = malloc((top + 1) * sizeof(*divsum)); uint pows[32] = {1, 0}; for (uint i = 0; i <= top; i++) divsum[i] = 1; // sieve // only sieve within lower half , the modification starts at 2*p for (uint p = 2; p+p <= top; p++) { if (divsum[p] > 1) { divsum[p] -= p;// subtract number itself from divisor sum ('proper') continue;} // p not prime uint x; // highest power of p we need //checking x <= top/y instead of x*y <= top to avoid overflow for (x = 1; pows[x - 1] <= top/p; x++) pows[x] = p*pows[x - 1]; //counter where n is not a*p with a = ?*p, useful for most p. //think of p>31 seldom divisions or p>sqrt(top) than no division is needed //n = 2*p, so the prime itself is left unchanged => k=p-1 uint k= p-1; for (uint n = p+p; n <= top; n += p) { uint s=1+pows[1]; k--; // search the right power only if needed if ( k==0) { for (uint i = 2; i < x && !(n%pows[i]); s += pows[i++]); k = p; } divsum[n] *= s; } } //now correct the upper half for (uint p = (top >> 1)+1; p <= top; p++) { if (divsum[p] > 1){ divsum[p] -= p;} } uint cnt = 0; for (uint a = 1; a <= top; a++) { uint b = divsum[a]; if (b > a && b <= top && divsum[b] == a){ printf("%u %u\n", a, b); cnt++;} } printf("\nTop %u count : %u\n",top,cnt); return 0; }  {{out}}  % ./a.out 20000 220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416 Top 20000 count : 8 % ./a.out 524000000 .. 475838415 514823985 491373104 511419856 509379344 523679536 Top 524000000 count : 442 real 0m16.285s user 0m16.156s  ## C++  #include <vector> #include <unordered_map> #include <iostream> int main() { std::vector<int> alreadyDiscovered; std::unordered_map<int, int> divsumMap; int count = 0; for (int N = 1; N <= 20000; ++N) { int divSumN = 0; for (int i = 1; i <= N / 2; ++i) { if (fmod(N, i) == 0) { divSumN += i; } } // populate map of integers to the sum of their proper divisors if (divSumN != 1) // do not include primes divsumMap[N] = divSumN; for (std::unordered_map<int, int>::iterator it = divsumMap.begin(); it != divsumMap.end(); ++it) { int M = it->first; int divSumM = it->second; int divSumN = divsumMap[N]; if (N != M && divSumM == N && divSumN == M) { // do not print duplicate pairs if (std::find(alreadyDiscovered.begin(), alreadyDiscovered.end(), N) != alreadyDiscovered.end()) break; std::cout << "[" << M << ", " << N << "]" << std::endl; alreadyDiscovered.push_back(M); alreadyDiscovered.push_back(N); count++; } } } std::cout << count << " amicable pairs discovered" << std::endl; }  {{out}}  [220, 284] [1184, 1210] [2620, 2924] [5020, 5564] [6232, 6368] [10744, 10856] [12285, 14595] [17296, 18416] 8 amicable pairs discovered  ## C# using System; using System.Collections.Generic; using System.Linq; namespace RosettaCode.AmicablePairs { internal static class Program { private const int Limit = 20000; private static void Main() { foreach (var pair in GetPairs(Limit)) { Console.WriteLine("{0} {1}", pair.Item1, pair.Item2); } } private static IEnumerable<Tuple<int, int>> GetPairs(int max) { List<int> divsums = Enumerable.Range(0, max + 1).Select(i => ProperDivisors(i).Sum()).ToList(); for(int i=1; i<divsums.Count; i++) { int sum = divsums[i]; if(i < sum && sum <= divsums.Count && divsums[sum] == i) { yield return new Tuple<int, int>(i, sum); } } } private static IEnumerable<int> ProperDivisors(int number) { return Enumerable.Range(1, number / 2) .Where(divisor => number % divisor == 0); } } }  {{out}}  220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416  ## Clojure  (ns example (:gen-class)) (defn factors [n] " Find the proper factors of a number " (into (sorted-set) (mapcat (fn [x] (if (= x 1) [x] [x (/ n x)])) (filter #(zero? (rem n %)) (range 1 (inc (Math/sqrt n)))) ))) (def find-pairs (into #{} (for [n (range 2 20000) :let [f (factors n) ; Factors of n M (apply + f) ; Sum of factors g (factors M) ; Factors of sum N (apply + g)] ; Sum of Factors of sum :when (= n N) ; (sum(proDivs(N)) = M and sum(propDivs(M)) = N :when (not= M N)] ; N not-equal M (sorted-set n M)))) ; Found pair ;; Output Results (doseq [q find-pairs] (println q))  {{Out}}  #{220 284} #{6232 6368} #{1184 1210} #{5020 5564} #{2620 2924} #{12285 14595} #{17296 18416} #{10744 10856}  ## Common Lisp (let ((cache (make-hash-table))) (defun sum-proper-divisors (n) (or (gethash n cache) (setf (gethash n cache) (loop for x from 1 to (/ n 2) when (zerop (rem n x)) sum x))))) (defun amicable-pairs-up-to (n) (loop for x from 1 to n for sum-divs = (sum-proper-divisors x) when (and (< x sum-divs) (= x (sum-proper-divisors sum-divs))) collect (list x sum-divs))) (amicable-pairs-up-to 20000)  {{out}} ((220 284) (1184 1210) (2620 2924) (5020 5564) (6232 6368) (10744 10856) (12285 14595) (17296 18416))  ## Crystal  MX = 524_000_000 N = Math.sqrt(MX).to_u32 x = Array(Int32).new(MX+1, 1) (2..N).each { |i| p = i*i x[p] += i k = i+i+1 (p+i..MX).step(i) { |j| x[j] += k k += 1 } } (4..MX).each { |m| n = x[m] if n < m && n != 0 && m == x[n] puts "#{n} #{m}" end }  {{out}}  220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416 ...... ...... ....... ....... 426191535 514780497 475838415 514823985 509379344 523679536  ## D {{trans|Python}} void main() @safe /*@nogc*/ { import std.stdio, std.algorithm, std.range, std.typecons, std.array; immutable properDivs = (in uint n) pure nothrow @safe /*@nogc*/ => iota(1, (n + 1) / 2 + 1).filter!(x => n % x == 0); enum rangeMax = 20_000; auto n2d = iota(1, rangeMax + 1).map!(n => properDivs(n).sum); foreach (immutable n, immutable divSum; n2d.enumerate(1)) if (n < divSum && divSum <= rangeMax && n2d[divSum - 1] == n) writefln("Amicable pair: %d and %d with proper divisors:\n %s\n %s", n, divSum, properDivs(n), properDivs(divSum)); }  {{out}} Amicable pair: 220 and 284 with proper divisors: [1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110] [1, 2, 4, 71, 142] Amicable pair: 1184 and 1210 with proper divisors: [1, 2, 4, 8, 16, 32, 37, 74, 148, 296, 592] [1, 2, 5, 10, 11, 22, 55, 110, 121, 242, 605] Amicable pair: 2620 and 2924 with proper divisors: [1, 2, 4, 5, 10, 20, 131, 262, 524, 655, 1310] [1, 2, 4, 17, 34, 43, 68, 86, 172, 731, 1462] Amicable pair: 5020 and 5564 with proper divisors: [1, 2, 4, 5, 10, 20, 251, 502, 1004, 1255, 2510] [1, 2, 4, 13, 26, 52, 107, 214, 428, 1391, 2782] Amicable pair: 6232 and 6368 with proper divisors: [1, 2, 4, 8, 19, 38, 41, 76, 82, 152, 164, 328, 779, 1558, 3116] [1, 2, 4, 8, 16, 32, 199, 398, 796, 1592, 3184] Amicable pair: 10744 and 10856 with proper divisors: [1, 2, 4, 8, 17, 34, 68, 79, 136, 158, 316, 632, 1343, 2686, 5372] [1, 2, 4, 8, 23, 46, 59, 92, 118, 184, 236, 472, 1357, 2714, 5428] Amicable pair: 12285 and 14595 with proper divisors: [1, 3, 5, 7, 9, 13, 15, 21, 27, 35, 39, 45, 63, 65, 91, 105, 117, 135, 189, 195, 273, 315, 351, 455, 585, 819, 945, 1365, 1755, 2457, 4095] [1, 3, 5, 7, 15, 21, 35, 105, 139, 417, 695, 973, 2085, 2919, 4865] Amicable pair: 17296 and 18416 with proper divisors: [1, 2, 4, 8, 16, 23, 46, 47, 92, 94, 184, 188, 368, 376, 752, 1081, 2162, 4324, 8648] [1, 2, 4, 8, 16, 1151, 2302, 4604, 9208]  ## EchoLisp  ;; using (sum-divisors) from math.lib (lib 'math) (define (amicable N) (define n 0) (for/list ((m (in-range 2 N))) (set! n (sum-divisors m)) #:continue (>= n (* 1.5 m)) ;; assume n/m < 1.5 #:continue (<= n m) ;; prevent perfect numbers #:continue (!= (sum-divisors n) m) (cons m n))) (amicable 20000) → ((220 . 284) (1184 . 1210) (2620 . 2924) (5020 . 5564) (6232 . 6368) (10744 . 10856) (12285 . 14595) (17296 . 18416)) (amicable 1_000_000) ;; 42 pairs → (... (802725 . 863835) (879712 . 901424) (898216 . 980984) (947835 . 1125765) (998104 . 1043096))  ## Ela {{trans|Haskell}} open monad io number list divisors n = filter ((0 ==) << (n mod)) [1..(n div 2)] range = [1 .. 20000] divs = zip range$ map (sum << divisors) range
pairs = [(n, m) \\ (n, nd) <- divs, (m, md) <- divs | n < m && nd == m && md == n]

do putLn pairs ::: IO


{{out}}


[(220,284),(1184,1210),(2620,2924),(5020,5564),(6232,6368),(10744,10856),(12285,14595),(17296,18416)]


## Elena

{{trans|C#}} ELENA 4.1 :

import extensions;
import system'routines;
import system'math;

const int N = 20000;

extension op
{
ProperDivisors
= Range.new(1,self / 2).filterBy:(n => self.mod:n == 0);

get AmicablePairs()
{
var divsums := Range
.new(0, self + 1)
.selectBy:(i => i.ProperDivisors.summarize(new Integer()))
.toArray();

^ 1.repeatTill(divsums.Length)
.filterBy:(i)
{
var ii := i;

var sum := divsums[i];
^ (i < sum) && (sum < divsums.Length) && (divsums[sum] == i)
}
.selectBy:(i => new::{ Item1 = i; Item2 = divsums[i]; })
}
}

public program()
{
N.AmicablePairs.forEach:(pair)
{
console.printLine(pair.Item1, " ", pair.Item2)
}
}


{{out}}


220 284
1184 1210
2620 2924
5020 5564
6232 6368
10744 10856
12285 14595
17296 18416



=== Alternative variant using strong-typed closures ===

import extensions;
import system'routines'stex;
import system'math;
import system'collections;

const int N = 20000;

extension op : IntNumber
{
Enumerator<int> ProperDivisors
= new Range(1,self / 2).filterBy:(int n => self.mod:n == 0);

get AmicablePairs()
{
auto divsums := new List<int>(cast Enumerator<int>(new Range(0, self).selectBy:(int i => i.ProperDivisors.summarize(0))));

^ new Range(0, divsums.Length)
.filterBy:(int i)
{
auto sum := divsums[i];
^ (i < sum) && (sum < divsums.Length) && (divsums[sum] == i)
}
.selectBy:(int i => new Tuple<int,int>(i,divsums[i]));
}
}

public program()
{
N.AmicablePairs.forEach:(var Tuple<int,int> pair)
{
console.printLine(pair.Item1, " ", pair.Item2)
}
}


{{out}}


220 284
1184 1210
2620 2924
5020 5564
6232 6368
10744 10856
12285 14595
17296 18416



## Elixir

{{works with|Elixir|1.2}} With [[proper_divisors#Elixir]] in place:

defmodule Proper do
def divisors(1), do: []
def divisors(n), do: [1 | divisors(2,n,:math.sqrt(n))] |> Enum.sort

defp divisors(k,_n,q) when k>q, do: []
defp divisors(k,n,q) when rem(n,k)>0, do: divisors(k+1,n,q)
defp divisors(k,n,q) when k * k == n, do: [k | divisors(k+1,n,q)]
defp divisors(k,n,q)                , do: [k,div(n,k) | divisors(k+1,n,q)]
end

map = Map.new(1..20000, fn n -> {n, Proper.divisors(n) |> Enum.sum} end)
Enum.filter(map, fn {n,sum} -> map[sum] == n and n < sum end)
|> Enum.sort
|> Enum.each(fn {i,j} -> IO.puts "#{i} and #{j}" end)


{{out}}


220 and 284
1184 and 1210
2620 and 2924
5020 and 5564
6232 and 6368
10744 and 10856
12285 and 14595
17296 and 18416



## Erlang

===Erlang, slow=== Very slow solution. Same functions by and large as in proper divisors and co.


-module(properdivs).
-export([amicable/1,divs/1,sumdivs/1]).

amicable(Limit) -> amicable(Limit,[],3,2).

amicable(Limit,List,_Current,Acc) when Acc >= Limit -> List;
amicable(Limit,List,Current,Acc) when Current =< Acc/2  ->
amicable(Limit,List,Acc,Acc+1);
amicable(Limit,List,Current,Acc) ->
CS = sumdivs(Current),
AS = sumdivs(Acc),
if
CS == Acc andalso AS == Current andalso Acc =/= Current ->
io:format("A: ~w, B: ~w, ~nL: ~w~w~n",  [Current,Acc,divs(Current),divs(Acc)]),
NL = List ++ [{Current,Acc}],
amicable(Limit,NL,Acc+1,Acc+1);
true ->
amicable(Limit,List,Current-1,Acc) end.

divs(0) -> [];
divs(1) -> [];
divs(N) -> lists:sort(divisors(1,N)).

divisors(1,N) ->
[1] ++ divisors(2,N,math:sqrt(N)).

divisors(K,_N,Q) when K > Q -> [];
divisors(K,N,_Q) when N rem K =/= 0 ->
[] ++ divisors(K+1,N,math:sqrt(N));
divisors(K,N,_Q) when K * K  == N ->
[K] ++ divisors(K+1,N,math:sqrt(N));
divisors(K,N,_Q) ->
[K, N div K] ++ divisors(K+1,N,math:sqrt(N)).

sumdivs(N) -> lists:sum(divs(N)).



{{out}}


3> properdivs:amicable(20000).
A: 220, B: 284,
L: [1,2,4,5,10,11,20,22,44,55,110][1,2,4,71,142]
A: 1184, B: 1210,
L: [1,2,4,8,16,32,37,74,148,296,592][1,2,5,10,11,22,55,110,121,242,605]
A: 2620, B: 2924,
L: [1,2,4,5,10,20,131,262,524,655,1310][1,2,4,17,34,43,68,86,172,731,1462]
A: 5020, B: 5564,
L: [1,2,4,5,10,20,251,502,1004,1255,2510][1,2,4,13,26,52,107,214,428,1391,2782]
A: 6232, B: 6368,
L: [1,2,4,8,19,38,41,76,82,152,164,328,779,1558,3116][1,2,4,8,16,32,199,398,796,1592,3184]
A: 10744, B: 10856,
L: [1,2,4,8,17,34,68,79,136,158,316,632,1343,2686,5372][1,2,4,8,23,46,59,92,118,184,236,472,1357,2714,5428]
A: 12285, B: 14595,
L: [1,3,5,7,9,13,15,21,27,35,39,45,63,65,91,105,117,135,189,195,273,315,351,455,585,819,945,1365,1755,2457,4095][1,3,5,7,15,21,35,105,139,417,695,973,2085,2919,4865]
A: 17296, B: 18416,
L: [1,2,4,8,16,23,46,47,92,94,184,188,368,376,752,1081,2162,4324,8648][1,2,4,8,16,1151,2302,4604,9208]
[{220,284},
{1184,1210},
{2620,2924},
{5020,5564},
{6232,6368},
{10744,10856},
{12285,14595},
{17296,18416}]



===Erlang, faster=== This is lazy AND depends on the fun fact that we're not really identifying pairs. They just happen to order. Probably, this answer is false in some sense. But a good deal faster :) As above with the additional function.

[See the talk section '''amicable pairs, out of order''' for this Rosetta Code task.]


friendly(Limit) ->
List = [{X,properdivs:sumdivs(X)} || X <- lists:seq(3,Limit)],
Final = [ X ||
X <- lists:seq(3,Limit),
X == properdivs:sumdivs(proplists:get_value(X,List))
andalso X =/= proplists:get_value(X,List)],
io:format("L: ~w~n", [Final]).



{{output}}


45> properdivs:friendly(20000).
L: [220,284,1184,1210,2620,2924,5020,5564,6232,6368,10744,10856,12285,14595,17296,18416]
ok



We might answer a challenge by saying:


friendly(Limit) ->
List = [{X,properdivs:sumdivs(X)} || X <- lists:seq(3,Limit)],
Final = [ X || X <- lists:seq(3,Limit), X == properdivs:sumdivs(proplists:get_value(X,List))
andalso X =/= proplists:get_value(X,List)],
findfriendlies(Final,[]).

findfriendlies(List,Acc) when length(List) =< 0 -> Acc;
findfriendlies(List,Acc) ->
A = lists:nth(1,List),
AS = sumdivs(A),
B = lists:nth(2,List),
BS = sumdivs(B),
if
AS == B andalso BS == A ->
{_,BL} = lists:split(2,List),
findfriendlies(BL,Acc++[{A,B}]);
true -> false
end.



{{out}}


94>  properdivs:friendly(20000).
[{220,284},
{1184,1210},
{2620,2924},
{5020,5564},
{6232,6368},
{10744,10856},
{12285,14595},
{17296,18416}]



In either case, it's a lot faster than the recursion in my first example.

## ERRE

PROGRAM AMICABLE

CONST LIMIT=20000

PROCEDURE SUMPROP(NUM->M)
IF NUM<2 THEN M=0 EXIT PROCEDURE
SUM=1
ROOT=SQR(NUM)
FOR I=2 TO ROOT-1 DO
IF (NUM=I*INT(NUM/I)) THEN
SUM=SUM+I+NUM/I
END IF
IF (NUM=ROOT*INT(NUM/ROOT)) THEN
SUM=SUM+ROOT
END IF
END FOR
M=SUM
END PROCEDURE

BEGIN
PRINT(CHR$(12);) ! CLS PRINT("Amicable pairs < ";LIMIT) FOR N=1 TO LIMIT DO SUMPROP(N->M1) SUMPROP(M1->M2) IF (N=M2 AND N<M1) THEN PRINT(N,M1) END FOR END PROGRAM  {{out}} Amicable pairs < 20000 220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416  ## F#  [2..20000 - 1] |> List.map (fun n-> n, ([1..n/2] |> List.filter (fun x->n % x = 0) |> List.sum)) |> List.map (fun (a,b) ->if a<b then (a,b) else (b,a)) |> List.groupBy id |> List.map snd |> List.filter (List.length >> ((=) 2)) |> List.map List.head |> List.iter (printfn "%A")  {{out}}  (220, 284) (1184, 1210) (2620, 2924) (5020, 5564) (6232, 6368) (10744, 10856) (12285, 14595) (17296, 18416)  ## Factor This solution focuses on the language's namesake: factoring code into small words which are subsequently composed to form more powerful — yet just as simple — words. Using this approach, the final word naturally arrives at the solution. This is often referred to as the bottom-up approach, which is a way in which Factor (and other concatenative languages) commonly differs from other languages.  USING: grouping math.primes.factors math.ranges ; : pdivs ( n -- seq ) divisors but-last ; : dsum ( n -- sum ) pdivs sum ; : dsum= ( n m -- ? ) dsum = ; : both-dsum= ( n m -- ? ) [ dsum= ] [ swap dsum= ] 2bi and ; : amicable? ( n m -- ? ) [ both-dsum= ] [ = not ] 2bi and ; : drange ( -- seq ) 2 20000 [a,b) ; : dsums ( -- seq ) drange [ dsum ] map ; : is-am?-seq ( -- seq ) dsums drange [ amicable? ] 2map ; : am-nums ( -- seq ) t is-am?-seq indices ; : am-nums-c ( -- seq ) am-nums [ 2 + ] map ; : am-pairs ( -- seq ) am-nums-c 2 group ; : print-am ( -- ) am-pairs [ >array . ] each ; print-am  {{out}}  { 220 284 } { 1184 1210 } { 2620 2924 } { 5020 5564 } { 6232 6368 } { 10744 10856 } { 12285 14595 } { 17296 18416 }  ## Fortran This version uses some latter-day facilities such as array assignment that could be replaced by an ordinary DO-loop, as could the FOR ALL statement that for two adds two to every second element, for three adds three to every third, etc. Each FORALL statement applies its DO-given increment to all the selected array elements potentially in any order or even simultaneously. Likewise, the "MODULE" protocol could be abandoned, which would mean that the KNOWNSUM array would have to be declared COMMON for access across routines - or the whole re-written as a single mainline. And if the PARAMETER statements were replaced appropriately, this source could be compiled using Fortran 77. Output: Perfect!! 6 Perfect!! 28 Amicable! 220 284 Perfect!! 496 Amicable! 1184 1210 Amicable! 2620 2924 Amicable! 5020 5564 Amicable! 6232 6368 Perfect!! 8128 Amicable! 10744 10856 Amicable! 12285 14595 Amicable! 17296 18416  MODULE FACTORSTUFF !This protocol evades the need for multiple parameters, or COMMON, or one shapeless main line... Concocted by R.N.McLean, MMXV. INTEGER LOTS,ILIMIT !Some bounds. PARAMETER (ILIMIT = 2147483647) !Computer arithmetic is not with real numbers. PARAMETER (LOTS = 22000) !Nor is computer storage infinite. INTEGER KNOWNSUM(LOTS) !Calculate these once as multiple references are expected. CONTAINS !Assistants. INTEGER FUNCTION SUMF(N) !Sum of the proper divisors of N. INTEGER N !The number in question. INTEGER S,F,F2,INC,BOOST !Assistants. IF (N.LE.LOTS) THEN !If we're within reach, SUMF = KNOWNSUM(N) !The result is to hand. ELSE !Otherwise, some on-the-spot effort ensues. Could use SUMF in place of S, but some compilers have been confused by such usage. S = 1 !1 is always a factor of N, but N is deemed not. F = 1 !Prepare a crude search for factors. INC = 1 !One by plodding one. IF (MOD(N,2) .EQ. 1) INC = 2!Ah, but an odd number cannot have an even number as a divisor. 1 F = F + INC !So half the time we can doubleplod. F2 = F*F !Up to F2 < N rather than F < SQRT(N) and worries over inexact arithmetic. IF (F2 .LT. N) THEN !F2 = N handled below. IF (MOD(N,F) .EQ. 0) THEN !Does F divide N? BOOST = F + N/F !Yes. The divisor and its counterpart. IF (S .GT. ILIMIT - BOOST) GO TO 666 !Would their augmentation cause an overflow? S = S + BOOST !No, so count in the two divisors just discovered. END IF !So much for a divisor discovered. GO TO 1 !Try for another. END IF !So much for the horde. IF (F2 .EQ. N) THEN !Special case: N may be a perfect square, not necessarily of a prime number. IF (S .GT. ILIMIT - F) GO TO 666 !It is. And it too might cause overflow. S = S + F !But if not, count F once only. END IF !All done. SUMF = S !This is the result. END IF !Whichever way obtained, RETURN !Done. Cannot calculate the sum, because it exceeds the integer limit. 666 SUMF = -666 !An expression of dismay that the caller will notice. END FUNCTION SUMF !Alternatively, find the prime factors, and combine them... SUBROUTINE PREPARESUMF !Initialise the KNOWNSUM array. Convert the Sieve of Eratoshenes to have each slot contain the sum of the proper divisors of its slot number. Changes to instead count the number of factors, or prime factors, etc. would be simple enough. INTEGER F !A factor for numbers such as 2F, 3F, 4F, 5F, ... KNOWNSUM(1) = 0 !Proper divisors of N do not include N. KNOWNSUM(2:LOTS) = 1 !So, although 1 is a proper divisor of all N, 1 is excluded for itself. DO F = 2,LOTS/2 !Step through all the possible divisors of numbers not exceeding LOTS. FOR ALL(I = F + F:LOTS:F) KNOWNSUM(I) = KNOWNSUM(I) + F !And augment each corresponding slot. END DO !Different divisors can hit the same slot. For instance, 6 by 2 and also by 3. END SUBROUTINE PREPARESUMF !Could alternatively generate all products of prime numbers. END MODULE FACTORSTUFF !Enough assistants. PROGRAM AMICABLE !Seek N such that SumF(SumF(N)) = N, for N up to 20,000. USE FACTORSTUFF !This should help. INTEGER I,N !Steppers. INTEGER S1,S2 !Sums of factors. CALL PREPARESUMF !Values for every N up to the search limit will be called for at least once. c WRITE (6,66) (I,KNOWNSUM(I), I = 1,48) c 66 FORMAT (10(I3,":",I5,"|")) DO N = 2,20000 !Step through the specified search space. S1 = SUMF(N) !Only even numbers appear in the results, but check every one anyway. IF (S1 .EQ. N) THEN !Catch a tight loop. WRITE (6,*) "Perfect!!",N !Self amicable! Would otherwise appear as Amicable! n,n. ELSE IF (S1 .GT. N) THEN !Look for a pair going upwards only. S2 = SUMF(S1) !Since otherwise each would appear twice. IF (S2.EQ.N) WRITE (6,*) "Amicable!",N,S1 !Aha! END IF !So much for that candidate. END DO !On to the next. END !Done.  ## FreeBASIC ### using Mod  ' FreeBASIC v1.05.0 win64 Function SumProperDivisors(number As Integer) As Integer If number < 2 Then Return 0 Dim sum As Integer = 0 For i As Integer = 1 To number \ 2 If number Mod i = 0 Then sum += i Next Return sum End Function Dim As Integer n, f Dim As Integer sum(19999) For n = 1 To 19999 sum(n) = SumProperDivisors(n) Next Print "The pairs of amicable numbers below 20,000 are :" Print For n = 1 To 19998 ' f = SumProperDivisors(n) f = sum(n) If f <= n OrElse f < 1 OrElse f > 19999 Then Continue For If f = sum(n) AndAlso n = sum(f) Then Print Using "#####"; n; Print " and "; Using "#####"; sum(n) End If Next Print Print "Press any key to exit the program" Sleep End  {{out}}  The pairs of amicable numbers below 20,000 are : 220 and 284 1184 and 1210 2620 and 2924 5020 and 5564 6232 and 6368 10744 and 10856 12285 and 14595 17296 and 18416  ===using "Sieve of Erathosthenes" style=== ' version 04-10-2016 ' compile with: fbc -s console ' replaced the function with 2 FOR NEXT loops #Define max 20000 ' test for pairs below max #Define max_1 max -1 Dim As String u_str = String(Len(Str(max))+1,"#") Dim As UInteger n, f Dim Shared As UInteger sum(max_1) For n = 2 To max_1 sum(n) = 1 Next For n = 2 To max_1 \ 2 For f = n * 2 To max_1 Step n sum(f) += n Next Next Print Print Using " The pairs of amicable numbers below" & u_str & ", are :"; max Print For n = 1 To max_1 -1 f = Sum(n) If f <= n OrElse f > max Then Continue For If f = sum(n) AndAlso n = sum(f) Then Print Using u_str & " and" & u_str ; n; f End If Next ' empty keyboard buffer While Inkey <> "" : Wend Print : Print : Print " Hit any key to end program" Sleep End   The pairs of amicable numbers below 20,000 are : 220 and 284 1184 and 1210 2620 and 2924 5020 and 5564 6232 and 6368 10744 and 10856 12285 and 14595 17296 and 18416  ## Frink This example uses Frink's built-in efficient factorization algorithms. It can work for arbitrarily large numbers.  n = 1 seen = new set do { n = n + 1 if seen.contains[n] next sum = sum[allFactors[n, true, false, false]] if sum != n and sum[allFactors[sum, true, false, false]] == n { println["$n, $sum"] seen.put[sum] } } while n <= 20000  {{out}}  220, 284 1184, 1210 2620, 2924 5020, 5564 6232, 6368 10744, 10856 12285, 14595 17296, 18416  ## Futhark {{output?}} This program is much too parallel and manifests all the pairs, which requires a giant amount of memory. fun divisors(n: int): []int = filter (fn x => n%x == 0) (map (1+) (iota (n/2))) fun amicable((n: int, nd: int), (m: int, md: int)): bool = n < m && nd == m && md == n fun getPair (divs: [upper](int, int)) (flat_i: int): ((int,int), (int,int)) = let i = flat_i / upper let j = flat_i % upper in unsafe (divs[i], divs[j]) fun main(upper: int): [][2]int = let range = map (1+) (iota upper) let divs = zip range (map (fn n => reduce (+) 0 (divisors n)) range) let amicable = filter amicable (map (getPair divs) (iota (upper*upper))) in map (fn (np,mp) => [#1 np, #1 mp]) amicable  ## GFA Basic <lang> OPENW 1 CLEARW 1 ' DIM f%(20001) ! sum of proper factors for each n FOR i%=1 TO 20000 f%(i%)=@sum_proper_divisors(i%) NEXT i% ' look for pairs FOR i%=1 TO 20000 FOR j%=i%+1 TO 20000 IF f%(i%)=j% AND i%=f%(j%) PRINT "Amicable pair ";i%;" ";j% ENDIF NEXT j% NEXT i% ' PRINT PRINT "-- found all amicable pairs" ~INP(2) CLOSEW 1 ' ' Compute the sum of proper divisors of given number ' FUNCTION sum_proper_divisors(n%) LOCAL i%,sum%,root% ' IF n%>1 ! n% must be 2 or larger sum%=1 ! start with 1 root%=SQR(n%) ! note that root% is an integer ' check possible factors, up to sqrt FOR i%=2 TO root% IF n% MOD i%=0 sum%=sum%+i% ! i% is a factor IF i%*i%<>n% ! check i% is not actual square root of n% sum%=sum%+n%/i% ! so n%/i% will also be a factor ENDIF ENDIF NEXT i% ENDIF RETURN sum% ENDFUNC  Output is:  Amicable pair: 220 284 Amicable pair: 1184 1210 Amicable pair: 2620 2924 Amicable pair: 5020 5564 Amicable pair: 6232 6368 Amicable pair: 10744 10856 Amicable pair: 12285 14595 Amicable pair: 17296 18416 -- found all amicable pairs  ## Go  package main import "fmt" import "math" func main() { var i int var a [200001]int a[0]=0 for i=1;i<=20000;i++{ a[i]=pfac_sum(i) } fmt.Println("The amicable pairs are:") for i=1;i<=20000;i++{ if (i==a[a[i]])&&(i<a[i]){ fmt.Printf("%d , %d\n",i,a[i]) } } } func pfac_sum(i int) int { var p,sum=1,0 for p=1;p<=i/2;p++{ x := float64(i) y := float64(p) if math.Mod(x,y)==0{ sum= sum+p } } return sum }  Output:  The amicable pairs are: 220 , 284 1184 , 1210 2620 , 2924 5020 , 5564 6232 , 6368 10744 , 10856 12285 , 14595 17296 , 18416  ## Haskell divisors :: (Integral a) => a -> [a] divisors n = filter ((0 ==) . (n mod)) [1 .. (n div 2)] main :: IO () main = do let range = [1 .. 20000 :: Int] divs = zip range$ map (sum . divisors) range
pairs = [(n, m) | (n, nd) <- divs, (m, md) <- divs,
n < m, nd == m, md == n]
print pairs


{{out}}

[(220,284),(1184,1210),(2620,2924),(5020,5564),(6232,6368),(10744,10856),(12285,14595),(17296,18416)]


Or, deriving proper divisors above the square root as cofactors (for better performance)

import Data.Bool (bool)

amicablePairsUpTo :: Int -> [(Int, Int)]
amicablePairsUpTo n =
let sigma = sum . properDivisors
in [1 .. n] >>=
(\x ->
let y = sigma x
in bool [] [(x, y)] (x < y && x == sigma y))

properDivisors
:: Integral a
=> a -> [a]
properDivisors n =
let root = (floor . sqrt) (fromIntegral n :: Double)
lows = filter ((0 ==) . rem n) [1 .. root]
in init $lows ++ drop (bool 0 1 (root * root == n)) (reverse (quot n <$> lows))

main :: IO ()
main = mapM_ print amicablePairsUpTo 20000  {{Out}} (220,284) (1184,1210) (2620,2924) (5020,5564) (6232,6368) (10744,10856) (12285,14595) (17296,18416)  ## J [[Proper divisors#J|Proper Divisor implementation]]: factors=: [: /:~@, */&>@{@((^ i.@>:)&.>/)@q:~&__ properDivisors=: factors -. -.&1  Amicable pairs:  1 + 0 20000 #: I. ,(</~@i.@# * (* |:))(=/ +/@properDivisors@>) 1 + i.20000 220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416  ## Java {{works with|Java|8}} import java.util.Map; import java.util.function.Function; import java.util.stream.Collectors; import java.util.stream.LongStream; public class AmicablePairs { public static void main(String[] args) { int limit = 20_000; Map<Long, Long> map = LongStream.rangeClosed(1, limit) .parallel() .boxed() .collect(Collectors.toMap(Function.identity(), AmicablePairs::properDivsSum)); LongStream.rangeClosed(1, limit) .forEach(n -> { long m = map.get(n); if (m > n && m <= limit && map.get(m) == n) System.out.printf("%s %s %n", n, m); }); } public static Long properDivsSum(long n) { return LongStream.rangeClosed(1, (n + 1) / 2).filter(i -> n % i == 0).sum(); } }  220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416  ## JavaScript ### ES5 (function (max) { // Proper divisors function properDivisors(n) { if (n < 2) return []; else { var rRoot = Math.sqrt(n), intRoot = Math.floor(rRoot), lows = range(1, intRoot).filter(function (x) { return (n % x) === 0; }); return lows.concat(lows.slice(1).map(function (x) { return n / x; }).reverse().slice((rRoot === intRoot) | 0)); } } // [m..n] function range(m, n) { var a = Array(n - m + 1), i = n + 1; while (i--) a[i - 1] = i; return a; } // Filter an array of proper divisor sums, // reading the array index as a function of N (N-1) // and the sum of proper divisors as a potential M var pairs = range(1, max).map(function (x) { return properDivisors(x).reduce(function (a, d) { return a + d; }, 0) }).reduce(function (a, m, i, lst) { var n = i + 1; return (m > n) && lst[m - 1] === n ? a.concat([[n, m]]) : a; }, []); // [[a]] -> bool -> s -> s function wikiTable(lstRows, blnHeaderRow, strStyle) { return '{| class="wikitable" ' + ( strStyle ? 'style="' + strStyle + '"' : '' ) + lstRows.map(function (lstRow, iRow) { var strDelim = ((blnHeaderRow && !iRow) ? '!' : '|'); return '\n|-\n' + strDelim + ' ' + lstRow.map(function (v) { return typeof v === 'undefined' ? ' ' : v; }).join(' ' + strDelim + strDelim + ' '); }).join('') + '\n|}'; } return wikiTable( [['N', 'M']].concat(pairs), true, 'text-align:center' ) + '\n\n' + JSON.stringify(pairs); })(20000);  {{out}} {| class="wikitable" style="text-align:center" |- ! N !! M |- | 220 || 284 |- | 1184 || 1210 |- | 2620 || 2924 |- | 5020 || 5564 |- | 6232 || 6368 |- | 10744 || 10856 |- | 12285 || 14595 |- | 17296 || 18416 |} [[220,284],[1184,1210],[2620,2924],[5020,5564], [6232,6368],[10744,10856],[12285,14595],[17296,18416]]  ### ES6 (() => { 'use strict'; // amicablePairsUpTo :: Int -> [(Int, Int)] const amicablePairsUpTo = n => { const sigma = compose(sum, properDivisors); return enumFromTo(1)(n).flatMap(x => { const y = sigma(x); return x < y && x === sigma(y) ? ([ [x, y] ]) : []; }); }; // properDivisors :: Int -> [Int] const properDivisors = n => { const rRoot = Math.sqrt(n), intRoot = Math.floor(rRoot), lows = enumFromTo(1)(intRoot) .filter(x => 0 === (n % x)); return lows.concat(lows.map(x => n / x) .reverse() .slice((rRoot === intRoot) | 0, -1)); }; // TEST ----------------------------------------------- // main :: IO () const main = () => console.log(unlines( amicablePairsUpTo(20000).map(JSON.stringify) )); // GENERIC FUNCTIONS ---------------------------------- // compose (<<<) :: (b -> c) -> (a -> b) -> a -> c const compose = (...fs) => x => fs.reduceRight((a, f) => f(a), x); // enumFromTo :: Int -> Int -> [Int] const enumFromTo = m => n => Array.from({ length: 1 + n - m }, (_, i) => m + i); // sum :: [Num] -> Num const sum = xs => xs.reduce((a, x) => a + x, 0); // unlines :: [String] -> String const unlines = xs => xs.join('\n'); // MAIN --- return main(); })();  {{Out}} [220,284] [1184,1210] [2620,2924] [5020,5564] [6232,6368] [10744,10856] [12285,14595] [17296,18416]  ## jq # unordered def proper_divisors: . asn
| if $n > 1 then 1, (sqrt|floor as$s
| range(2; $s+1) as$i
| if ($n %$i) == 0 then $i, (if$i * $i ==$n then empty else ($n /$i) end)
else empty
end)
else empty
end;

def addup(stream): reduce stream as $i (0; . +$i);

(reduce range(0; n+1) as $n ( []; . + [$n | addup(proper_divisors)] )) as $listing | range(1;n+1) as$j
| range(1;$j) as$k
| if $listing[$j] == $k and$listing[$k] ==$j
then "\($k) and \($j) are amicable"
else empty
end ;



{{out}}

$jq -c -n -f amicable_pairs.jq 220 and 284 are amicable 1184 and 1210 are amicable 2620 and 2924 are amicable 5020 and 5564 are amicable 6232 and 6368 are amicable 10744 and 10856 are amicable 12285 and 14595 are amicable 17296 and 18416 are amicable  ## Julia Given factor, it is not necessary to calculate the individual divisors to compute their sum. See [[Abundant,_deficient_and_perfect_number_classifications#Julia|Abundant, deficient and perfect number classifications]] for the details. '''Functions'''  function pcontrib(p::Int64, a::Int64) n = one(p) pcon = one(p) for i in 1:a n *= p pcon += n end return pcon end function divisorsum(n::Int64) dsum = one(n) for (p, a) in factor(n) dsum *= pcontrib(p, a) end dsum -= n end  pcontrib is a good candidate for [[Memoization]] should the performance of divisorsum become an issue. '''Main''' It is safe to exclude primes from consideration; their proper divisor sum is always 1. Also, this code uses a minor trick to ensure that none of the numbers identified are above the limit. All numbers in the range are checked for an amicable partner, but the pair is cataloged only when the greater member is reached.  using Primes const L = 2*10^4 acnt = 0 println("Amicable pairs not greater than ", L) for i in 2:L !isprime(i) || continue j = divisorsum(i) j < i && divisorsum(j) == i || continue acnt += 1 println(@sprintf("%4d", acnt), " => ", j, ", ", i) end  {{out}}  Amicable pairs not greater than 20000 1 => 220, 284 2 => 1184, 1210 3 => 2620, 2924 4 => 5020, 5564 5 => 6232, 6368 6 => 10744, 10856 7 => 12285, 14595 8 => 17296, 18416  ## K  propdivs:{1+&0=x!'1+!x%2} (8,2)#v@&{(x=+/propdivs[a])&~x=a:+/propdivs[x]}' v:1+!20000 (220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416)  ## Kotlin // version 1.1 fun sumProperDivisors(n: Int): Int { if (n < 2) return 0 return (1..n / 2).filter{ (n % it) == 0 }.sum() } fun main(args: Array<String>) { val sum = IntArray(20000, { sumProperDivisors(it) } ) println("The pairs of amicable numbers below 20,000 are:\n") for(n in 2..19998) { val m = sum[n] if (m > n && m < 20000 && n == sum[m]) { println(n.toString().padStart(5) + " and " + m.toString().padStart(5)) } } }  {{out}}  The pairs of amicable numbers below 20,000 are: 220 and 284 1184 and 1210 2620 and 2924 5020 and 5564 6232 and 6368 10744 and 10856 12285 and 14595 17296 and 18416  ## Lua 0.02 of a second in 16 lines of code. The vital trick is to just set m to the sum of n's proper divisors each time. That way you only have to test the reverse, dividing your run time by half the loop limit (ie. 10,000)! function sumDivs (n) local sum = 1 for d = 2, math.sqrt(n) do if n % d == 0 then sum = sum + d sum = sum + n / d end end return sum end for n = 2, 20000 do m = sumDivs(n) if m > n then if sumDivs(m) == n then print(n, m) end end end  {{out}}  220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416  ## Maple {{output?}}  with(NumberTheory): pairs:=[]; for i from 1 to 20000 do for j from i+1 to 20000 do sum1:=SumOfDivisors(j)-j; sum2:=SumOfDivisors(i)-i; if sum1=i and sum2=j and i<>j then pairs:=[op(pairs),[i,j]]; printf("%a", pairs); end if; end do; end do; pairs;  ## Matlab  function amicable tic N=2:1:20000; aN=[]; N(isprime(N))=[]; %erase prime numbers I=1; a=N(1); b=sum(pd(a)); while length(N)>1 if a==b %erase perfect numbers; N(N==a)=[]; a=N(1); b=sum(pd(a)); elseif b<a %the first member of an amicable pair is abundant not defective N(N==a)=[]; a=N(1); b=sum(pd(a)); elseif ~ismember(b,N) %the other member was previously erased N(N==a)=[]; a=N(1); b=sum(pd(a)); else c=sum(pd(b)); if a==c aN(I,:)=[I a b]; I=I+1; N(N==b)=[]; else if ~ismember(c,N) %the other member was previously erased N(N==b)=[]; end end N(N==a)=[]; a=N(1); b=sum(pd(a)); clear c end end disp(array2table(aN,'Variablenames',{'N','Amicable1','Amicable2'})) toc end function D=pd(x) K=1:ceil(x/2); D=K(~(rem(x, K))); end  {{out}}  N Amicable1 Amicable2 _ _________ _________ 1 220 284 2 1184 1210 3 2620 2924 4 5020 5564 5 6232 6368 6 10744 10856 7 12285 14595 8 17296 18416 Elapsed time is 8.958720 seconds.  =={{header|Mathematica}} / {{header|Wolfram Language}}== amicableQ[n_] := Module[{sum = Total[Most@Divisors@n]}, sum != n && n == Total[Most@Divisors@sum]] Grid@Partition[Cases[Range[4, 20000], _?(amicableQ@# &)], 2]  {{out}}  220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416  ## Nim Being a novice, I submitted my code to the Nim community for review and received much feedback and advice. They were instrumental in fine-tuning this code for style and readability, I can't thank them enough.  from math import sqrt const N = 524_000_000.int32 proc sumProperDivisors(someNum: int32, chk4less: bool): int32 = result = 1 let maxPD = sqrt(someNum.float).int32 let offset = someNum mod 2 for divNum in countup(2 + offset, maxPD, 1 + offset): if someNum mod divNum == 0: result += divNum + someNum div divNum if chk4less and result >= someNum: return 0 for n in countdown(N, 2): let m = sumProperDivisors(n, true) if m != 0 and n == sumProperDivisors(m, false): echo$n, " ", $m  {{out}}  523679536 509379344 511419856 491373104 514823985 475838415 ...... ...... ..... ..... 18416 17296 14595 12285 10856 10744 6368 6232 5564 5020 2924 2620 1210 1184 284 220  Total number of pairs is 442, on my machine the code takes ~389 minutes to run. Here's a second version that uses a large amount of memory but runs in 2m32seconds. Again, thanks to the Nim community  from math import sqrt const N = 524_000_000.int32 var x = newSeq[int32](N+1) for i in 2..sqrt(N.float).int32: var p = i*i x[p] += i var j = i + i while (p += i; p <= N): j.inc x[p] += j for m in 4..N: let n = x[m] + 1 if n < m and n != 0 and m == x[n] + 1: echo n, " ", m  {{out}}  220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416 ..... ..... ...... ...... 426191535 514780497 475838415 514823985 509379344 523679536  =={{header|Oberon-2}}==  MODULE AmicablePairs; IMPORT Out; CONST max = 20000; VAR i,j: INTEGER; pd: ARRAY max + 1 OF LONGINT; PROCEDURE ProperDivisorsSum(n: LONGINT): LONGINT; VAR i,sum: LONGINT; BEGIN sum := 0; IF n > 1 THEN INC(sum,1);i := 2; WHILE (i < n) DO IF (n MOD i) = 0 THEN INC(sum,i) END; INC(i) END END; RETURN sum END ProperDivisorsSum; BEGIN FOR i := 0 TO max DO pd[i] := ProperDivisorsSum(i) END; FOR i := 2 TO max DO FOR j := i + 1 TO max DO IF (pd[i] = j) & (pd[j] = i) THEN Out.Char('[');Out.Int(i,0);Out.Char(',');Out.Int(j,0);Out.Char("]");Out.Ln END END END END AmicablePairs.  {{Out}}  [220,284] [1184,1210] [2620,2924] [5020,5564] [6232,6368] [10744,10856] [12285,14595] [17296,18416]  ## Oforth Using properDivs implementation tasks without optimization (calculating proper divisors sum without returning a list for instance) : import: mapping Integer method: properDivs -- [] #[ self swap mod 0 == ] self 2 / seq filter ; : amicables | i j | Array new 20000 loop: i [ i properDivs sum dup ->j i <= if continue then j properDivs sum i <> if continue then [ i, j ] over add ] ;  {{out}}  amicables . [[220, 284], [1184, 1210], [2620, 2924], [5020, 5564], [6232, 6368], [10744, 10856], [12285, 14595], [17296, 18416]]  ## PARI/GP for(x=1,20000,my(y=sigma(x)-x); if(y>x && x == sigma(y)-y,print(x" "y)))  {{out}} 220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416  ## Pascal ### Direct approach {{works with|Turbo Pascal}}{{works with| Free Pascal}} This version mutates the Sieve of Eratoshenes from striking out factors into summing factors. The Pascal source compiles with Turbo Pascal (7, patched to avoid the zero divide problem for cpu speeds better than ~150MHz) except that the array limit is too large: 15,000 works but does not reach 20,000. The Free Pascal compiler however can handle an array of 20,000 elements. Because the sum of factors of N can exceed N an ad-hoc SumF procedure is provided, thus the search could continue past the table limit, but at a cost in calculation time. Output is Chasing Chains of Sums of Factors of Numbers. Perfect!! 6, Perfect!! 28, Amicable! 220,284, Perfect!! 496, Amicable! 1184,1210, Amicable! 2620,2924, Amicable! 5020,5564, Amicable! 6232,6368, Perfect!! 8128, Amicable! 10744,10856, Amicable! 12285,14595, Sociable: 12496,14288,15472,14536,14264, Sociable: 14316,19116,31704,47616,83328,177792,295488,629072,589786,294896,358336,418904,366556,274924,275444,243760,376736,381028,285778,152990,122410,97946,48976,45946,22976,22744,19916,17716, Amicable! 17296,18416, Source file:  Program SumOfFactors; uses crt; {Perpetrated by R.N.McLean, December MCMXCV} //{$DEFINE ShowOverflow}
{$IFDEF FPC} {$MODE DELPHI}//tested with lots = 524*1000*1000 takes 75 secs generating KnownSum
{$ENDIF} var outf: text; const Limit = 2147483647; const lots = 20000; {This should be much bigger, but problems apply.} var KnownSum: array[1..lots] of longint; Function SumF(N: Longint): Longint; var f,f2,s,ulp: longint; Begin if n <= lots then SumF:=KnownSum[N] {Hurrah!} else begin {This is really crude...} s:=1; {1 is always a factor, but N is not.} f:=2; f2:=f*f; while f2 < N do begin if N mod f = 0 then begin {We have a divisor, and its friend.} ulp:=f + (N div f); if s > Limit - ulp then begin SumF:=-666; exit; end; s:=s + ulp; end; f:=f + 1; f2:=f*f; end; if f2 = N then {A perfect square gets its factor in once only.} if s <= Limit - f then s:=s + f else begin SumF:=-667; exit; end; SumF:=s; end; End; var i,j,l,sf,fs: LongInt; const enuff = 666; {Only so much sociability.} var trail: array[0..enuff] of longint; BEGIN ClrScr; WriteLn('Chasing Chains of Sums of Factors of Numbers.'); for i:=1 to lots do KnownSum[i]:=1; {Sigh. KnownSum:=1;} {start summing every divisor } for i:=2 to lots do begin j:=i + i; While j <= lots do {Sigh. For j:=i + i:Lots:i do KnownSum[j]:=KnownSum[j] + i;} begin KnownSum[j]:=KnownSum[j] + i; j:=j + i; end; end; {Enough preparation.} Assign(outf,'Factors.txt'); ReWrite(Outf); WriteLn(Outf,'Chasing Chains of Sums of Factors of Numbers.'); for i:=2 to lots do {Search.} begin l:=0; sf:=SumF(i); while (sf > i) and (l < enuff) do begin l:=l + 1; trail[l]:=sf; sf:=SumF(sf); end; if l >= enuff then writeln('Rope ran out! ',i); {$IFDEF ShowOverflow}
if sf < 0 then writeln('Overflow with ',i);
{$ENDIF} if i = sf then {A loop?} begin {Yes. Reveal its members.} trail[0]:=i; {The first.} if l = 0 then write('Perfect!! ') else if l = 1 then write('Amicable! ') else write('Sociable: '); for j:=0 to l do Write(Trail[j],','); WriteLn; if l = 0 then write(outf,'Perfect!! ') else if l = 1 then write(outf,'Amicable! ') else write(outf,'Sociable: '); for j:=0 to l do write(outf,Trail[j],','); WriteLn(outf); end; end; Close (outf); END.  ### More expansive. a "normal" Version. Nearly fast as perl using nTheory. program AmicablePairs; {$IFDEF FPC}
{$MODE DELPHI} {$H+}
{$ELSE} {$APPTYPE CONSOLE}
{$ENDIF} uses sysutils; const MAX = 20000; //MAX = 20*1000*1000; type tValue = LongWord; tpValue = ^tValue; tPower = array[0..31] of tValue; tIndex = record idxI, idxS : Uint64; end; var Indices : array[0..511] of tIndex; //primes up to 65536 enough until 2^32 primes : array[0..6542] of tValue; procedure InitPrimes; // sieve of erathosthenes without multiples of 2 type tSieve = array[0..(65536-1) div 2] of char; var ESieve : ^tSieve; idx,i,j,p : LongINt; Begin new(ESieve); fillchar(ESieve^[0],SizeOF(tSieve),#1); primes[0] := 2; idx := 1; //sieving j := 1; p := 2*j+1; repeat if Esieve^[j] = #1 then begin i := (2*j+2)*j;// i := (sqr(p) -1) div 2; if i > High(tSieve) then BREAK; repeat ESIeve^[i] := #0; inc(i,p); until i > High(tSieve); end; inc(j); inc(p,2); until j >High(tSieve); //collecting For i := 1 to High(tSieve) do IF Esieve^[i] = #1 then Begin primes[idx] := 2*i+1; inc(idx); IF idx>High(primes) then BREAK; end; dispose(Esieve); end; procedure Su_append(n,factor:tValue;var su:string); var q,p : tValue; begin p := 0; repeat q := n div factor; IF q*factor<>n then Break; inc(p); n := q; until false; IF p > 0 then IF p= 1 then su:= su+IntToStr(factor)+'*' else su:= su+IntToStr(factor)+'^'+IntToStr(p)+'*'; end; procedure ProperDivs(n: Uint64); //output of prime factorization var su : string; primNo : tValue; p:tValue; begin str(n:8,su); su:= su +' ['; primNo := 0; p := primes[0]; repeat Su_Append(n,p,su); inc(primNo); p := primes[primNo]; until (p=0) OR (p*p >= n); p := n; Su_Append(n,p,su); su[length(su)] := ']'; writeln(su); end; procedure AmPairOutput(cnt:tValue); var i : tValue; r_max,r_min,r : double; begin r_max := 1.0; r_min := 16.0; For i := 0 to cnt-1 do with Indices[i] do begin r := IdxS/IDxI; writeln(i+1:4,IdxI:16,IDxS:16,' ratio ',r:10:7); IF r < 1 then begin writeln(i); readln; halt; end; if r_max < r then r_max := r else if r_min > r then r_min := r; IF cnt < 20 then begin ProperDivs(IdxI); ProperDivs(IdxS); end; end; writeln(' min ratio ',r_min:12:10); writeln(' max ratio ',r_max:12:10); end; procedure SumOFProperDiv(n: tValue;var SumOfProperDivs:tValue); // calculated by prime factorization var i,q, primNo, Prime,pot : tValue; SumOfDivs: tValue; begin i := N; SumOfDivs := 1; primNo := 0; Prime := Primes[0]; q := i DIV Prime; repeat if q*Prime = i then Begin pot := 1; repeat i := q; q := i div Prime; Pot := Pot * Prime+1; until q*Prime <> i; SumOfDivs := SumOfDivs * pot; end; Inc(primNo); Prime := Primes[primNo]; q := i DIV Prime; {check if i already prime} if Prime > q then begin prime := i; q := 1; end; until i = 1; SumOfProperDivs := SumOfDivs - N; end; function Check:tValue; const //going backwards DIV23 : array[0..5] of byte = //== 5,4,3,2,1,0 (1,0,0,0,1,0); var i,s,k,n : tValue; idx : nativeInt; begin n := 0; idx := 3; For i := 2 to MAX do begin //must be divisble by 2 or 3 ( n < High(tValue) < 1e14 ) IF DIV23[idx] = 0 then begin SumOFProperDiv(i,s); //only 24.7...% IF s>i then Begin SumOFProperDiv(s,k); IF k = i then begin With indices[n] do begin idxI := i; idxS := s; end; inc(n); end; end; end; dec(idx); IF idx < 0 then idx := high(DIV23); end; result := n; end; var T2,T1: TDatetime; APcnt: tValue; begin InitPrimes; T1:= time; APCnt:= Check; T2:= time; AmPairOutput(APCnt); writeln('Time to find amicable pairs ',FormatDateTime('HH:NN:SS.ZZZ' ,T2-T1)); {$IFNDEF UNIX} readln;{$ENDIF} end.  Output  1 220 284 ratio 1.2909091 220 [2^2*5*11*220] 284 [2^2*284] 2 1184 1210 ratio 1.0219595 1184 [2^5*1184] 1210 [2*5*11^2*1210] 3 2620 2924 ratio 1.1160305 2620 [2^2*5*2620] 2924 [2^2*17*43*2924] 4 5020 5564 ratio 1.1083665 5020 [2^2*5*5020] 5564 [2^2*13*5564] 5 6232 6368 ratio 1.0218228 6232 [2^3*19*41*6232] 6368 [2^5*6368] 6 10744 10856 ratio 1.0104244 10744 [2^3*17*79*10744] 10856 [2^3*23*59*10856] 7 12285 14595 ratio 1.1880342 12285 [3^3*5*7*13*12285] 14595 [3*5*7*14595] 8 17296 18416 ratio 1.0647549 17296 [2^4*23*47*17296] 18416 [2^4*18416]  ### Alternative about 25-times faster. This will not output the amicable number unless both! numbers are under the given limit. So there will be differences to "Table of n, a(n) for n=1..39374" https://oeis.org/A002025/b002025.txt Up to 524'000'000 the pairs found are only correct by number up to no. 437 460122410 and only 442 out of 455 are found, because some pairs exceed the limit. The limits of the ratio between the numbers of the amicable pair up to 1E14 are, based on b002025.txt:  No. lower upper 31447 52326552030976 52326637800704 ratio 1.0000016 52326552030976 [2^8*563*6079*59723] 52326637800704 [2^8*797*1439*178223] 38336 92371445691525 154378742017851 ratio 1.6712821 92371445691525 [3^2*5^2*7^2*11*13^2*23*29^2*233] 154378742017851 [3^2*13^2*53*337*5682671]  The distance check is being corrected, the lower number is now not limited. The used method is not useful for very high limits. n = p[1]^a[1]p[2]^a[2]...p[l]^a[l] sum of divisors(n) = s(n) = (p[1]^(a[1]+1) -1) / (p[1] -1) * ... * (p[l]^(a[l]+1) -1) / (p[l] -1) with p[k]^(a[k]+1) -1) / (p[k] -1) = sum (i= [1..a[k]])(p[k]^i) Using "Sieve of Erathosthenes"-style program AmicPair; {find amicable pairs in a limited region 2..MAX beware that >both< numbers must be smaller than MAX there are 455 amicable pairs up to 524*1000*1000 correct up to #437 460122410 } //optimized for freepascal 2.6.4 32-Bit {$IFDEF FPC}
{$MODE DELPHI} {$OPTIMIZATION ON,peephole,cse,asmcse,regvar}
{$CODEALIGN loop=1,proc=8} {$ELSE}
{$APPTYPE CONSOLE} {$ENDIF}

uses
sysutils;

type
tValue = LongWord;
tpValue = ^tValue;
tDivSum = array[0..0] of tValue;// evil, but dynamic arrays are slower
tpDivSum = ^tDivSum;
tPower = array[0..31] of tValue;
tIndex = record
idxI,
idxS : tValue;
end;
var
power,
PowerFac     : tPower;
ds           : array of tValue;
Indices      : array[0..511] of tIndex;
DivSumField  : tpDivSum;
MAX : tValue;

procedure Init;
var
i : LongInt;
begin
DivSumField[0]:= 0;
For i := 1 to MAX do
DivSumField[i]:= 1;
end;

procedure ProperDivs(n: tValue);
//Only for output, normally a factorication would do
var
su,so : string;
i,q : tValue;
begin
su:= '1';
so:= '';
i := 2;
while i*i <= n do
begin
q := n div i;
IF q*i -n = 0 then
begin
su:= su+','+IntToStr(i);
IF q <> i then
so:= ','+IntToStr(q)+so;
end;
inc(i);
end;
writeln('  [',su+so,']');
end;

procedure AmPairOutput(cnt:tValue);
var
i : tValue;
r : double;
begin
r := 1.0;
For i := 0 to cnt-1 do
with Indices[i] do
begin
writeln(i+1:4,IdxI:12,IDxS:12,' ratio ',IdxS/IDxI:10:7);
if r < IdxS/IDxI then
r := IdxS/IDxI;
IF cnt < 20 then
begin
ProperDivs(IdxI);
ProperDivs(IdxS);
end;
end;
writeln(' max ratio ',r:10:4);
end;

function Check:tValue;
var
i,s,n : tValue;
begin
n := 0;
For i := 1 to MAX do
begin
//s = sum of proper divs (I)  == sum of divs (I) - I
s := DivSumField^[i];
IF (s <=MAX) AND (s>i) AND (DivSumField^[s]= i)then
begin
With indices[n] do
begin
idxI := i;
idxS := s;
end;
inc(n);
end;
end;
result := n;
end;

Procedure CalcPotfactor(prim:tValue);
//PowerFac[k] = (prim^(k+1)-1)/(prim-1) == Sum (i=0..k) prim^i
var
k: tValue;
Pot,       //== prim^k
PFac : Int64;
begin
Pot := prim;
PFac := 1;
For k := 0 to High(PowerFac) do
begin
PFac := PFac+Pot;
IF (POT > MAX) then
BREAK;
PowerFac[k] := PFac;
Pot := Pot*prim;
end;
end;

procedure InitPW(prim:tValue);
begin
fillchar(power,SizeOf(power),#0);
CalcPotfactor(prim);
end;

function NextPotCnt(p: tValue):tValue;
//return the first power <> 0
//power == n to base prim
var
i : tValue;
begin
result := 0;
repeat
i := power[result];
Inc(i);
IF i < p then
BREAK
else
begin
i := 0;
power[result]  := 0;
inc(result);
end;
until false;
power[result] := i;
end;

procedure Sieve(prim: tValue);
var
actNumber,idx : tValue;
begin
//sieve with "small" primes
while prim*prim <= MAX do
begin
InitPW(prim);
Begin
//actNumber = actual number = n*prim
actNumber := prim;
idx := prim;
while actNumber <= MAX do
begin
dec(idx);
IF idx > 0 then
DivSumField^[actNumber] *= PowerFac[0]
else
Begin
DivSumField^[actNumber] *= PowerFac[NextPotCnt(prim)+1];
idx := Prim;
end;
inc(actNumber,prim);
end;
end;
//next prime
repeat
inc(prim);
until DivSumField^[prim]= 1;//(DivSumField[prim] = 1);
end;

//sieve with "big" primes, only one factor is possible
while 2*prim <= MAX do
begin
InitPW(prim);
Begin
actNumber := prim;
idx := PowerFac[0];
while actNumber <= MAX do
begin
DivSumField^[actNumber] *= idx;
inc(actNumber,prim);
end;
end;
repeat
inc(prim);
until DivSumField^[prim]= 1;
end;

For idx := 2 to MAX do
dec(DivSumField^[idx],idx);
end;

var
T2,T1,T0: TDatetime;
APcnt: tValue;
i: NativeInt;
begin
MAX := 20000;
IF  ParamCount > 0 then
MAX := StrToInt(ParamStr(1));
setlength(ds,MAX);
DivSumField := @ds[0];
T0:= time;
For i := 1 to 1 do
Begin
Init;
Sieve(2);
end;
T1:= time;

APCnt := Check;
T2:= time;
AmPairOutput(APCnt);
writeln(APCnt,' amicable pairs til ',MAX);
writeln('Time to calc sum of divs    ',FormatDateTime('HH:NN:SS.ZZZ' ,T1-T0));
writeln('Time to find amicable pairs ',FormatDateTime('HH:NN:SS.ZZZ' ,T2-T1));
setlength(ds,0);
{$IFNDEF UNIX} readln; {$ENDIF}
end.


output


220       284
[1,2,4,5,10,11,20,22,44,55,110]
[1,2,4,71,142]

1184      1210
[1,2,4,8,16,32,37,74,148,296,592]
[1,2,5,10,11,22,55,110,121,242,605]

2620      2924
[1,2,4,5,10,20,131,262,524,655,1310]
[1,2,4,17,34,43,68,86,172,731,1462]

5020      5564
[1,2,4,5,10,20,251,502,1004,1255,2510]
[1,2,4,13,26,52,107,214,428,1391,2782]

6232      6368
[1,2,4,8,19,38,41,76,82,152,164,328,779,1558,3116]
[1,2,4,8,16,32,199,398,796,1592,3184]

10744     10856
[1,2,4,8,17,34,68,79,136,158,316,632,1343,2686,5372]
[1,2,4,8,23,46,59,92,118,184,236,472,1357,2714,5428]

12285     14595
[1,3,5,7,9,13,15,21,27,35,39,45,63,65,91,105,117,135,189,195,273,315,351,455,585,819,945,1365,1755,2457,4095]
[1,3,5,7,15,21,35,105,139,417,695,973,2085,2919,4865]

17296     18416
[1,2,4,8,16,23,46,47,92,94,184,188,368,376,752,1081,2162,4324,8648]
[1,2,4,8,16,1151,2302,4604,9208]

8 amicable numbers up to 20000
00:00:00.000

.... Test with 524*1000*1000 Linux32, FPC 3.0.1, i4330 3.5 Ghz //Win32 swaps first to allocate 2 GB )
440   475838415   514823985 ratio  1.0819303
441   491373104   511419856 ratio  1.0407974
442   509379344   523679536 ratio  1.0280738
max ratio     1.3537
442 amicable pairs til 524000000
Time to calc sum of divs    00:00:12.601
Time to find amicable pairs 00:00:02.557



## Perl

Not particularly clever, but instant for this example, and does up to 20 million in 11 seconds. {{libheader|ntheory}}

use ntheory qw/divisor_sum/;
for my $x (1..20000) { my$y = divisor_sum($x)-$x;
say "$x$y" if $y >$x && $x == divisor_sum($y)-$y; }  {{out}} 220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416  ## Perl 6 {{Works with|Rakudo|2019.03.1}} sub propdivsum (\x) { my @l = 1 if x > 1; (2 .. x.sqrt.floor).map: -> \d { unless x % d { @l.push: d; my \y = x div d; @l.push: y if y != d } } sum @l } (1..20000).race.map: ->$i {
my $j = propdivsum($i);
say "$i$j" if $j >$i and $i == propdivsum($j);
}


{{out}}

220 284
1184 1210
2620 2924
5020 5564
6232 6368
10744 10856
12285 14595
17296 18416


## Phix

integer n
for m=1 to 20000 do
n = sum(factors(m,-1))
if m<n and m=sum(factors(n,-1)) then ?{m,n} end if
end for


{{out}}


{220,284}
{1184,1210}
{2620,2924}
{5020,5564}
{6232,6368}
{10744,10856}
{12285,14595}
{17296,18416}



## PicoLisp

(de accud (Var Key)
(if (assoc Key (val Var))
(con @ (inc (cdr @)))
(push Var (cons Key 1)) )
Key )
(de **sum (L)
(let S 1
(for I (cdr L)
(inc 'S (** (car L) I)) )
S ) )
(de factor-sum (N)
(if (=1 N)
0
(let
(R NIL
D 2
L (1 2 2 . (4 2 4 2 4 6 2 6 .))
M (sqrt N)
N1 N
S 1 )
(while (>= M D)
(if (=0 (% N1 D))
(setq M
(sqrt (setq N1 (/ N1 (accud 'R D)))) )
(inc 'D (pop 'L)) ) )
(accud 'R N1)
(for I R
(setq S (* S (**sum I))) )
(- S N) ) ) )
(bench
(for I 20000
(let X (factor-sum I)
(and
(< I X)
(= I (factor-sum X))
(println I X) ) ) ) )


{{output}}


220 284
1184 1210
2620 2924
5020 5564
6232 6368
10744 10856
12285 14595
17296 18416
0.101 sec



## PL/I

{{trans|REXX}}

*process source xref;
ami: Proc Options(main);
p9a=time();
Dcl (p9a,p9b,p9c) Pic'(9)9';
Dcl sumpd(20000) Bin Fixed(31);
Dcl pd(300) Bin Fixed(31);
Dcl npd     Bin Fixed(31);
Dcl (x,y)   Bin Fixed(31);

Do x=1 To 20000;
Call proper_divisors(x,pd,npd);
sumpd(x)=sum(pd,npd);
End;
p9b=time();
Put Edit('sum(pd) computed in',(p9b-p9a)/1000,' seconds elapsed')
(Skip,col(7),a,f(6,3),a);

Do x=1 To 20000;
Do y=x+1 To 20000;
If y=sumpd(x) &
x=sumpd(y) Then
Put Edit(x,y,' found after ',elapsed(),' seconds')
(Skip,2(f(6)),a,f(6,3),a);
End;
End;
Put Edit(elapsed(),' seconds total search time')(Skip,f(6,3),a);

proper_divisors: Proc(n,pd,npd);
Dcl (n,pd(300),npd) Bin Fixed(31);
Dcl (d,delta)       Bin Fixed(31);
npd=0;
If n>1 Then Do;
If mod(n,2)=1 Then  /* odd number  */
delta=2;
Else                /* even number */
delta=1;
Do d=1 To n/2 By delta;
If mod(n,d)=0 Then Do;
npd+=1;
pd(npd)=d;
End;
End;
End;
End;

sum: Proc(pd,npd) Returns(Bin Fixed(31));
Dcl (pd(300),npd) Bin Fixed(31);
Dcl sum Bin Fixed(31) Init(0);
Dcl i   Bin Fixed(31);
Do i=1 To npd;
sum+=pd(i);
End;
Return(sum);
End;

elapsed: Proc Returns(Dec Fixed(6,3));
p9c=time();
Return((p9c-p9b)/1000);
End;
End;


{{out}}

      sum(pd) computed in 0.510 seconds elapsed
220   284 found after  0.010 seconds
1184  1210 found after  0.060 seconds
2620  2924 found after  0.110 seconds
5020  5564 found after  0.210 seconds
6232  6368 found after  0.260 seconds
10744 10856 found after  2.110 seconds
12285 14595 found after  2.150 seconds
17296 18416 found after  2.240 seconds
2.250 seconds total search time


## PowerShell

{{works with|PowerShell|2}}


function Get-ProperDivisorSum ( [int]$N ) {$Sum = 1
If ( $N -gt 3 ) {$SqrtN = [math]::Sqrt( $N ) ForEach ($Divisor1 in 2..$SqrtN ) {$Divisor2 = $N /$Divisor1
If ( $Divisor2 -is [int] ) {$Sum += $Divisor1 +$Divisor2 }
}
If ( $SqrtN -is [int] ) {$Sum -= $SqrtN } } return$Sum
}

function Get-AmicablePairs ( $N = 300 ) { ForEach ($X in 1..$N ) {$Sum = Get-ProperDivisorSum $X If ($Sum -gt $X -and$X -eq ( Get-ProperDivisorSum $Sum ) ) { "$X, Sum" } } } Get-AmicablePairs 20000  {{out}}  220, 284 1184, 1210 2620, 2924 5020, 5564 6232, 6368 10744, 10856 12285, 14595 17296, 18416  ## Prolog {{works with|SWI-Prolog 7}} With some guidance from other solutions here: divisor(N, Divisor) :- UpperBound is round(sqrt(N)), between(1, UpperBound, D), 0 is N mod D, ( Divisor = D ; LargerDivisor is N/D, LargerDivisor =\= D, Divisor = LargerDivisor ). proper_divisor(N, D) :- divisor(N, D), D =\= N. assoc_num_divsSum_in_range(Low, High, Assoc) :- findall( Num-DivSum, ( between(Low, High, Num), aggregate_all( sum(D), proper_divisor(Num, D), DivSum )), Pairs ), list_to_assoc(Pairs, Assoc). get_amicable_pair(Assoc, M-N) :- gen_assoc(M, Assoc, N), M < N, get_assoc(N, Assoc, M). amicable_pairs_under_20000(Pairs) :- assoc_num_divsSum_in_range(1,20000, Assoc), findall(P, get_amicable_pair(Assoc, P), Pairs).  Output: ?- amicable_pairs_under_20000(R). R = [220-284, 1184-1210, 2620-2924, 5020-5564, 6232-6368, 10744-10856, 12285-14595, 17296-18416].  ## PureBasic  EnableExplicit Procedure.i SumProperDivisors(Number) If Number < 2 : ProcedureReturn 0 : EndIf Protected i, sum = 0 For i = 1 To Number / 2 If Number % i = 0 sum + i EndIf Next ProcedureReturn sum EndProcedure Define n, f Define Dim sum(19999) If OpenConsole() For n = 1 To 19999 sum(n) = SumProperDivisors(n) Next PrintN("The pairs of amicable numbers below 20,000 are : ") PrintN("") For n = 1 To 19998 f = sum(n) If f <= n Or f < 1 Or f > 19999 : Continue : EndIf If f = sum(n) And n = sum(f) PrintN(RSet(Str(n),5) + " and " + RSet(Str(sum(n)), 5)) EndIf Next PrintN("") PrintN("Press any key to close the console") Repeat: Delay(10) : Until Inkey() <> "" CloseConsole() EndIf  {{out}}  The pairs of amicable numbers below 20,000 are : 220 and 284 1184 and 1210 2620 and 2924 5020 and 5564 6232 and 6368 10744 and 10856 12285 and 14595 17296 and 18416  ## Python Importing [[Proper_divisors#Python:_From_prime_factors|Proper divisors from prime factors]]: from proper_divisors import proper_divs def amicable(rangemax=20000): n2divsum = {n: sum(proper_divs(n)) for n in range(1, rangemax + 1)} for num, divsum in n2divsum.items(): if num < divsum and divsum <= rangemax and n2divsum[divsum] == num: yield num, divsum if __name__ == '__main__': for num, divsum in amicable(): print('Amicable pair: %i and %i With proper divisors:\n %r\n %r' % (num, divsum, sorted(proper_divs(num)), sorted(proper_divs(divsum))))  {{out}} Amicable pair: 220 and 284 With proper divisors: [1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110] [1, 2, 4, 71, 142] Amicable pair: 1184 and 1210 With proper divisors: [1, 2, 4, 8, 16, 32, 37, 74, 148, 296, 592] [1, 2, 5, 10, 11, 22, 55, 110, 121, 242, 605] Amicable pair: 2620 and 2924 With proper divisors: [1, 2, 4, 5, 10, 20, 131, 262, 524, 655, 1310] [1, 2, 4, 17, 34, 43, 68, 86, 172, 731, 1462] Amicable pair: 5020 and 5564 With proper divisors: [1, 2, 4, 5, 10, 20, 251, 502, 1004, 1255, 2510] [1, 2, 4, 13, 26, 52, 107, 214, 428, 1391, 2782] Amicable pair: 6232 and 6368 With proper divisors: [1, 2, 4, 8, 19, 38, 41, 76, 82, 152, 164, 328, 779, 1558, 3116] [1, 2, 4, 8, 16, 32, 199, 398, 796, 1592, 3184] Amicable pair: 10744 and 10856 With proper divisors: [1, 2, 4, 8, 17, 34, 68, 79, 136, 158, 316, 632, 1343, 2686, 5372] [1, 2, 4, 8, 23, 46, 59, 92, 118, 184, 236, 472, 1357, 2714, 5428] Amicable pair: 12285 and 14595 With proper divisors: [1, 3, 5, 7, 9, 13, 15, 21, 27, 35, 39, 45, 63, 65, 91, 105, 117, 135, 189, 195, 273, 315, 351, 455, 585, 819, 945, 1365, 1755, 2457, 4095] [1, 3, 5, 7, 15, 21, 35, 105, 139, 417, 695, 973, 2085, 2919, 4865] Amicable pair: 17296 and 18416 With proper divisors: [1, 2, 4, 8, 16, 23, 46, 47, 92, 94, 184, 188, 368, 376, 752, 1081, 2162, 4324, 8648] [1, 2, 4, 8, 16, 1151, 2302, 4604, 9208]  Or, supplying our own '''properDivisors''' function, and defining the harvest in terms of a generic '''concatMap''': '''Amicable pairs''' from itertools import chain from math import sqrt # amicablePairsUpTo :: Int -> [(Int, Int)] def amicablePairsUpTo(n): '''List of all amicable pairs of integers below n. ''' sigma = compose(sum)(properDivisors) def amicable(x): y = sigma(x) return [(x, y)] if (x < y and x == sigma(y)) else [] return concatMap(amicable)( enumFromTo(1)(n) ) # TEST ---------------------------------------------------- # main :: IO () def main(): '''Amicable pairs of integers up to 20000''' for x in amicablePairsUpTo(20000): print(x) # GENERIC ------------------------------------------------- # compose (<<<) :: (b -> c) -> (a -> b) -> a -> c def compose(g): '''Right to left function composition.''' return lambda f: lambda x: g(f(x)) # concatMap :: (a -> [b]) -> [a] -> [b] def concatMap(f): '''A concatenated list or string over which a function f has been mapped. The list monad can be derived by using an (a -> [b]) function which wraps its output in a list (using an empty list to represent computational failure). ''' return lambda xs: (''.join if isinstance(xs, str) else list)( chain.from_iterable(map(f, xs)) ) # enumFromTo :: Int -> Int -> [Int] def enumFromTo(m): '''Enumeration of integer values [m..n]''' def go(n): return list(range(m, 1 + n)) return lambda n: go(n) # properDivisors :: Int -> [Int] def properDivisors(n): '''Positive divisors of n, excluding n itself''' root_ = sqrt(n) intRoot = int(root_) blnSqr = root_ == intRoot lows = [x for x in range(1, 1 + intRoot) if 0 == n % x] return lows + [ n // x for x in reversed( lows[1:-1] if blnSqr else lows[1:] ) ] # MAIN --- if __name__ == '__main__': main()  {{Out}} (220, 284) (1184, 1210) (2620, 2924) (5020, 5564) (6232, 6368) (10744, 10856) (12285, 14595) (17296, 18416)  ## R  divisors <- function (n) { Filter( function (m) 0 == n %% m, 1:(n/2) ) } table = sapply(1:19999, function (n) sum(divisors(n)) ) for (n in 1:19999) { m = table[n] if ((m > n) && (m < 20000) && (n == table[m])) cat(n, " ", m, "\n") }  {{out}}  220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416  ## Racket With [[Proper_divisors#Racket]] in place: #lang racket (require "proper-divisors.rkt") (define SCOPE 20000) (define P (let ((P-v (vector))) (λ (n) (set! P-v (fold-divisors P-v n 0 +)) (vector-ref P-v n)))) ;; returns #f if not an amicable number, amicable pairing otherwise (define (amicable? n) (define m (P n)) (define m-sod (P m)) (and (= m-sod n) (< m n) ; each pair exactly once, also eliminates perfect numbers m)) (void (amicable? SCOPE)) ; prime the memoisation (for* ((n (in-range 1 (add1 SCOPE))) (m (in-value (amicable? n))) #:when m) (printf #<<EOS amicable pair: ~a, ~a ~a: divisors: ~a ~a: divisors: ~a EOS n m n (proper-divisors n) m (proper-divisors m)))  {{out}} amicable pair: 284, 220 284: divisors: (1 2 4 71 142) 220: divisors: (1 2 4 5 10 11 20 22 44 55 110) amicable pair: 1210, 1184 1210: divisors: (1 2 5 10 11 22 55 110 121 242 605) 1184: divisors: (1 2 4 8 16 32 37 74 148 296 592) amicable pair: 2924, 2620 2924: divisors: (1 2 4 17 34 43 68 86 172 731 1462) 2620: divisors: (1 2 4 5 10 20 131 262 524 655 1310) amicable pair: 5564, 5020 5564: divisors: (1 2 4 13 26 52 107 214 428 1391 2782) 5020: divisors: (1 2 4 5 10 20 251 502 1004 1255 2510) amicable pair: 6368, 6232 6368: divisors: (1 2 4 8 16 32 199 398 796 1592 3184) 6232: divisors: (1 2 4 8 19 38 41 76 82 152 164 328 779 1558 3116) amicable pair: 10856, 10744 10856: divisors: (1 2 4 8 23 46 59 92 118 184 236 472 1357 2714 5428) 10744: divisors: (1 2 4 8 17 34 68 79 136 158 316 632 1343 2686 5372) amicable pair: 14595, 12285 14595: divisors: (1 3 5 7 15 21 35 105 139 417 695 973 2085 2919 4865) 12285: divisors: (1 3 5 7 9 13 15 21 27 35 39 45 63 65 91 105 117 135 189 195 273 315 351 455 585 819 945 1365 1755 2457 4095) amicable pair: 18416, 17296 18416: divisors: (1 2 4 8 16 1151 2302 4604 9208) 17296: divisors: (1 2 4 8 16 23 46 47 92 94 184 188 368 376 752 1081 2162 4324 8648)  ## REXX ===version 1, with factoring=== Call time 'R' Do x=1 To 20000 pd=proper_divisors(x) sumpd.x=sum(pd) End Say 'sum(pd) computed in' time('E') 'seconds' Call time 'R' Do x=1 To 20000 /* If x//1000=0 Then Say x time() */ Do y=x+1 To 20000 If y=sumpd.x &, x=sumpd.y Then Say x y 'found after' time('E') 'seconds' End End Say time('E') 'seconds total search time' Exit proper_divisors: Procedure Parse Arg n Pd='' If n=1 Then Return '' If n//2=1 Then /* odd number */ delta=2 Else /* even number */ delta=1 Do d=1 To n%2 By delta If n//d=0 Then pd=pd d End Return space(pd) sum: Procedure Parse Arg list sum=0 Do i=1 To words(list) sum=sum+word(list,i) End Return sum  {{out}} sum(pd) computed in 48.502000 seconds 220 284 found after 3.775000 seconds 1184 1210 found after 21.611000 seconds 2620 2924 found after 46.817000 seconds 5020 5564 found after 84.296000 seconds 6232 6368 found after 100.918000 seconds 10744 10856 found after 150.126000 seconds 12285 14595 found after 162.124000 seconds 17296 18416 found after 185.600000 seconds 188.836000 seconds total search time  ===version 2, using SIGMA function=== This REXX version allows the specification of the upper limit (for the searching of amicable pairs). Some optimization was incorporated by using a '''sigma''' function, which was a re-coded ''proper divisors'' (Pdivs) function, which was taken from the REXX language entry for Rosetta Code task ''integer factors''. Other optimizations were incorporated which took advantage of several well-known generalizations about amicable pairs. The generation/summation is about '''50''' times faster; searching is about '''100''' times faster. Time consumption note: for every doubling of '''H''' (the upper limit for searches), the time consumed triples. /*REXX program calculates and displays all amicable pairs up to a given number. */ parse arg H .; if H=='' | H=="," then H=20000 /*get optional arguments (high limit).*/ w=length(H) ; low=220 /*W: used for columnar output alignment*/ @.=. /* [↑] LOW is lowest amicable number. */ do k=low for H-low; _=sigma(k) /*generate sigma sums for a range of #s*/ if _>=low then @.k=_ /*only keep the pertinent sigma sums. */ end /*k*/ /* [↑] process a range of integers. */ #=0 /*number of amicable pairs found so far*/ do m=low to H; n=@.m /*start the search at the lowest number*/ if m==@.n then do /*If equal, might be an amicable number*/ if m==n then iterate /*Is this a perfect number? Then skip.*/ #=#+1 /*bump the amicable pair counter. */ say right(m,w) ' and ' right(n,w) " are an amicable pair." m=n /*start M (DO index) from N. */ end end /*m*/ say say # 'amicable pairs found up to' H /*display count of the amicable pairs. */ exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ sigma: procedure; parse arg x; od=x//2 /*use either EVEN or ODD integers. */ s=1 /*set initial sigma sum to unity. ___*/ do j=2+od by 1+od while j*j<x /*divide by all integers up to the √ x */ if x//j==0 then s=s + j + x%j /*add the two divisors to the sum. */ end /*j*/ /* [↑] % is REXX integer division. */ return s /*return the sum of the divisors. */  '''output''' when using the default input:  220 and 284 are an amicable pair. 1184 and 1210 are an amicable pair. 2620 and 2924 are an amicable pair. 5020 and 5564 are an amicable pair. 6232 and 6368 are an amicable pair. 10744 and 10856 are an amicable pair. 12285 and 14595 are an amicable pair. 17296 and 18416 are an amicable pair. 8 amicable pairs found up to 20000  ===version 3, SIGMA with limited searches=== This REXX version is optimized to take advantage of the lowest ending-single-digit amicable number, and also incorporates the search of amicable numbers into the generation of the sigmas of the integers. The optimization makes it about another 30% faster when searching for amicable numbers up to one million. /*REXX program calculates and displays all amicable pairs up to a given number. */ parse arg H .; if H=='' | H=="," then H=20000 /*get optional arguments (high limit).*/ w=length(H) ; low=220 /*W: used for columnar output alignment*/ x=220 34765731 6232 87633 284 12285 10856 36939357 6368 5684679 /*S minimums. */ do i=0 for 10;.i=word(x,i+1);  end /*i*/   /*minimum amicable #s for last dec dig.*/
#=0                                              /*number of amicable pairs found so far*/
@.=                                              /* [↑]  LOW is lowest amicable number. */
do k=low  for H-low                         /*generate sigma sums for a range of #s*/
parse var k  ''  -1  D                      /*obtain last decimal digit of   K.    */
if k<.D then iterate /*if no need to compute, then skip it. */ _=sigma(k) /*generate sigma sum for the number K.*/ @.k=_ /*only keep the pertinent sigma sums. */ if k==@._ then do /*is it a possible amicable number ? */ if _==k then iterate /*Is it a perfect number? Then skip it*/ #=#+1 /*bump the amicable pair counter. */ say right(_,w) ' and ' right(k,w) " are an amicable pair." end end /*k*/ /* [↑] process a range of integers. */ say say # 'amicable pairs found up to' H /*display the count of amicable pairs. */ exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ sigma: procedure; parse arg x; od=x//2 /*use either EVEN or ODD integers. */ s=1 /*set initial sigma sum to unity. ___*/ do j=2+od by 1+od while j*j<x /*divide by all integers up to the √ x */ if x//j==0 then s=s + j + x%j /*add the two divisors to the sum. */ end /*j*/ /* [↑] % is REXX integer division. */ return s /*return the sum of the divisors. */  '''output''' is the same as the 2nd REXX version. ===version 4, SIGMA using integer SQRT=== This REXX version is optimized to use the ''integer square root of X'' in the '''sigma''' function (instead of computing the square of '''J''' to see if that value exceeds '''X'''). The optimization makes it about another 20% faster when searching for amicable numbers up to one million. /*REXX program calculates and displays all amicable pairs up to a given number. */ parse arg H .; if H=='' | H=="," then H=20000 /*get optional arguments (high limit).*/ w=length(H) ; low=220 /*W: used for columnar output alignment*/ x=220 34765731 6232 87633 284 12285 10856 36939357 6368 5684679 /*S minimums. */ do i=0 for 10;.i=word(x,i+1);  end  /*i*/  /*minimum amicable #s for last dec dig.*/
#=0                                              /*number of amicable pairs found so far*/
@.=                                              /* [↑]  LOW is lowest amicable number. */
do k=low  for H-low                         /*generate sigma sums for a range of #s*/
parse var k  ''  -1  D                      /*obtain last decimal digit of   K.    */
if k<.D then iterate /*if no need to compute, then skip it. */ _=sigma(k) /*generate sigma sum for the number K.*/ @.k=_ /*only keep the pertinent sigma sums. */ if k==@._ then do /*is it a possible amicable number ? */ if _==k then iterate /*Is it a perfect number? Then skip it*/ #=#+1 /*bump the amicable pair counter. */ say right(_,w) ' and ' right(k,w) " are an amicable pair." end end /*k*/ /* [↑] process a range of integers. */ say say # 'amicable pairs found up to' H /*display the count of amicable pairs. */ exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ iSqrt: procedure; parse arg x; r=0; q=1; do while q<=x; q=q*4; end do while q>1; q=q%4; _=x-r-q; r=r%2; if _>=0 then do;x=_;r=r+q; end; end return r /*──────────────────────────────────────────────────────────────────────────────────────*/ sigma: procedure; parse arg x; od=x//2 /*use either EVEN or ODD integers. */ s=1 /*set initial sigma sum to unity. ___*/ do j=2+od by 1+od to iSqrt(x) /*divide by all integers up to the √ x */ if x//j==0 then s=s + j + x%j /*add the two divisors to the sum. */ end /*j*/ /* [↑] % is the REXX integer division.*/ return s /*return the sum of the divisors. */  '''output''' is the same as the 2nd REXX version. ===version 5, SIGMA (in-line code)=== This REXX version is optimized by bringing the functions in-line (which minimizes the overhead of invoking two internal functions), and it also pre-computes the powers of four (for the integer square root code). This method of coding has the disadvantage in that the code (logic) is less idiomatic and therefore less readable. The optimization makes it about another 15% faster when searching for amicable numbers up to one million. /*REXX program calculates and displays all amicable pairs up to a given number. */ parse arg H .; if H=='' | H=="," then H=20000 /*get optional arguments (high limit).*/ w=length(H) ; low=220 /*W: used for columnar output alignment*/ x=220 34765731 6232 87633 284 12285 10856 36939357 6368 5684679 /*S minimums.*/ do i=0 for 10;.i=word(x,i+1);  end  /*i*/  /*minimum amicable #s for last dec dig.*/
f.=0;   do p=0  until f.p>10**digits();  f.p=4**p;  end  /*p*/         /*calc. pows of 4*/
#=0                                              /*number of amicable pairs found so far*/
@.=                                              /* [↑]  LOW is lowest amicable number. */
do k=low  for H-low+1                       /*generate sigma sums for a range of #s*/
parse var k  ''  -1  D                      /*obtain last decimal digit of   K.    */
if k<$.D then iterate /*if no need to compute, then skip it. */ od=k//2 /*OD: set to unity if K is odd.*/ z=k; q=1; do p=0 while f.p<=z; q=f.p; end /*R will end up being the iSqrt of Z.*/ r=0; do while q>1; q=q%4; _=z-r-q; r=r%2; if _>=0 then do; z=_; r=r+q; end; end s=1 /*set initial sigma sum to unity. ___*/ do j=2+od by 1+od to r /*divide by all integers up to the √ K */ if k//j==0 then s=s+ j + k%j /*add the two divisors to the sum. */ end /*j*/ /* [↑] % is REXX integer division. */ @.k=s /*only keep the pertinent sigma sums. */ if k==@.s then do /*is it a possible amicable number ? */ if s==k then iterate /*Is it a perfect number? Then skip it*/ #=#+1 /*bump the amicable pair counter. */ say right(s,w) ' and ' right(k,w) " are an amicable pair." end end /*k*/ /* [↑] process a range of integers. */ say /*stick a fork in it, we're all done. */ say # 'amicable pairs found up to' H /*display the count of amicable pairs. */  '''output''' is the same as the 2nd REXX version. ## Ring  size = 18500 for n = 1 to size m = amicable(n) if m>n and amicable(m)=n see "" + n + " and " + m + nl ok next see "OK" + nl func amicable nr sum = 1 for d = 2 to sqrt(nr) if nr % d = 0 sum = sum + d sum = sum + nr / d ok next return sum  ## Ruby With [[proper_divisors#Ruby]] in place: h = {} (1..20_000).each{|n| h[n] = n.proper_divisors.sum } h.select{|k,v| h[v] == k && k < v}.each do |key,val| # k<v filters out doubles and perfects puts "#{key} and #{val}" end  {{out}}  220 and 284 1184 and 1210 2620 and 2924 5020 and 5564 6232 and 6368 10744 and 10856 12285 and 14595 17296 and 18416  ## Run BASIC size = 18500 for n = 1 to size m = amicable(n) if m > n and amicable(m) = n then print n ; " and " ; m next function amicable(nr) amicable = 1 for d = 2 to sqr(nr) if nr mod d = 0 then amicable = amicable + d + nr / d next end function   220 and 284 1184 and 1210 2620 and 2924 5020 and 5564 6232 and 6368 10744 and 10856 12285 and 14595 17296 and 18416  ## Rust fn sum_of_divisors(val: u32) -> u32 { (1..val/2+1).filter(|n| val % n == 0) .fold(0, |sum, n| sum + n) } fn main() { let iter = (1..20_000).map(|i| (i, sum_of_divisors(i))) .filter(|&(i, div_sum)| i > div_sum); for (i, sum1) in iter { if sum_of_divisors(sum1) == i { println!("{} {}", i, sum1); } } }  {{out}}  284 220 1210 1184 2924 2620 5564 5020 6368 6232 10856 10744 14595 12285 18416 17296  ## Scala def properDivisors(n: Int) = (1 to n/2).filter(i => n % i == 0) val divisorsSum = (1 to 20000).map(i => i -> properDivisors(i).sum).toMap val result = divisorsSum.filter(v => v._1 < v._2 && divisorsSum.get(v._2) == Some(v._1)) println( result mkString ", " )  {{out}} 5020 -> 5564, 220 -> 284, 6232 -> 6368, 17296 -> 18416, 2620 -> 2924, 10744 -> 10856, 12285 -> 14595, 1184 -> 1210  ## Scheme  (import (scheme base) (scheme inexact) (scheme write) (only (srfi 1) fold)) ;; return a list of the proper-divisors of n (define (proper-divisors n) (let ((root (sqrt n))) (let loop ((divisors (list 1)) (i 2)) (if (> i root) divisors (loop (if (zero? (modulo n i)) (if (= (square i) n) (cons i divisors) (append (list i (quotient n i)) divisors)) divisors) (+ 1 i)))))) (define (sum-proper-divisors n) (if (< n 2) 0 (fold + 0 (proper-divisors n)))) (define *max-n* 20000) ;; hold sums of proper divisors in a cache, to avoid recalculating (define *cache* (make-vector (+ 1 *max-n*))) (for-each (lambda (i) (vector-set! *cache* i (sum-proper-divisors i))) (iota *max-n* 1)) (define (amicable-pair? i j) (and (not (= i j)) (= i (vector-ref *cache* j)) (= j (vector-ref *cache* i)))) ;; double loop to *max-n*, displaying all amicable pairs (let loop-i ((i 1)) (when (<= i *max-n*) (let loop-j ((j i)) (when (<= j *max-n*) (when (amicable-pair? i j) (display (string-append "Amicable pair: " (number->string i) " " (number->string j))) (newline)) (loop-j (+ 1 j)))) (loop-i (+ 1 i))))  {{out}}  Amicable pair: 220 284 Amicable pair: 1184 1210 Amicable pair: 2620 2924 Amicable pair: 5020 5564 Amicable pair: 6232 6368 Amicable pair: 10744 10856 Amicable pair: 12285 14595 Amicable pair: 17296 18416  ## Sidef func propdivsum(n) { n.sigma - n } for i in (1..20000) { var j = propdivsum(i) say "#{i} #{j}" if (j>i && i==propdivsum(j)) }  {{out}}  220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416  ## Swift import func Darwin.sqrt func sqrt(x:Int) -> Int { return Int(sqrt(Double(x))) } func properDivs(n: Int) -> [Int] { if n == 1 { return [] } var result = [Int]() for div in filter (1...sqrt(n), { n %$0 == 0 }) {

result.append(div)

if n/div != div && n/div != n { result.append(n/div) }
}

return sorted(result)

}

func sumDivs(n:Int) -> Int {

struct Cache { static var sum = [Int:Int]() }

if let sum = Cache.sum[n] { return sum }

let sum = properDivs(n).reduce(0) { $0 +$1 }

Cache.sum[n] = sum

return sum
}

func amicable(n:Int, m:Int) -> Bool {

if n == m { return false }

if sumDivs(n) != m || sumDivs(m) != n { return false }

return true
}

var pairs = [(Int, Int)]()

for n in 1 ..< 20_000 {
for m in n+1 ... 20_000 {
if amicable(n, m) {
pairs.append(n, m)
println("\(n, m)")
}
}
}


### Alternative

import func Darwin.sqrt

func sqrt(x:Int) -> Int { return Int(sqrt(Double(x))) }

func sigma(n: Int) -> Int {

if n == 1 { return 0 }          // definition of aliquot sum

var result = 1
let root = sqrt(n)

for var div = 2; div <= root; ++div {
if n % div == 0 {
result += div + n/div
}

}
if root*root == n { result -= root }

return (result)
}

func amicables (upTo: Int) -> () {

var aliquot = Array(count: upTo+1, repeatedValue: 0)

for i in 1 ... upTo {           // fill lookup array
aliquot[i] = sigma(i)
}

for i in 1 ... upTo {
let a = aliquot[i]
if a > upTo {continue}      //second part of pair out-of-bounds

if a == i {continue}        //skip perfect numbers

if i == aliquot[a] {
print("\(i, a)")
aliquot[a] = upTo+1     //prevent second display of pair
}
}
}

amicables(20_000)


{{out}}

(220, 284)
(1184, 1210)
(2620, 2924)
(5020, 5564)
(6232, 6368)
(10744, 10856)
(12285, 14595)
(17296, 18416)



## tbas


dim sums(20000)

sub sum_proper_divisors(n)
dim sum = 0
dim i
if n > 1 then
for i = 1 to (n \ 2)
if n %% i = 0 then
sum = sum + i
end if
next
end if
return sum
end sub

dim i, j
for i = 1 to 20000
sums(i) = sum_proper_divisors(i)
for j = i-1 to 2 by -1
if sums(i) = j and sums(j) = i then
print "Amicable pair:";sums(i);"-";sums(j)
exit for
end if
next
next



>tbas amicable_pairs.bas
Amicable pair: 220 - 284
Amicable pair: 1184 - 1210
Amicable pair: 2620 - 2924
Amicable pair: 5020 - 5564
Amicable pair: 6232 - 6368
Amicable pair: 10744 - 10856
Amicable pair: 12285 - 14595
Amicable pair: 17296 - 18416



## Tcl

proc properDivisors {n} {
if {$n == 1} return set divs 1 set sum 1 for {set i 2} {$i*$i <=$n} {incr i} {
if {!($n %$i)} {
lappend divs $i incr sum$i
if {$i*$i < $n} { lappend divs [set d [expr {$n / $i}]] incr sum$d
}
}
}
return [list $sum$divs]
}

proc amicablePairs {limit} {
set result {}
set sums [set divs {{}}]
for {set n 1} {$n <$limit} {incr n} {
lassign [properDivisors $n] sum d lappend sums$sum
lappend divs [lsort -integer $d] } for {set n 1} {$n < $limit} {incr n} { set nsum [lindex$sums $n] for {set m 1} {$m < $n} {incr m} { if {$n==[lindex $sums$m] && $m==$nsum} {
lappend result $m$n [lindex $divs$m] [lindex $divs$n]
}
}
}
return $result } foreach {m n md nd} [amicablePairs 20000] { puts "$m and $n are an amicable pair with these proper divisors" puts "\t$m : $md" puts "\t$n : \$nd"
}


{{out}}


220 and 284 are an amicable pair with these proper divisors
220 : 1 2 4 5 10 11 20 22 44 55 110
284 : 1 2 4 71 142
1184 and 1210 are an amicable pair with these proper divisors
1184 : 1 2 4 8 16 32 37 74 148 296 592
1210 : 1 2 5 10 11 22 55 110 121 242 605
2620 and 2924 are an amicable pair with these proper divisors
2620 : 1 2 4 5 10 20 131 262 524 655 1310
2924 : 1 2 4 17 34 43 68 86 172 731 1462
5020 and 5564 are an amicable pair with these proper divisors
5020 : 1 2 4 5 10 20 251 502 1004 1255 2510
5564 : 1 2 4 13 26 52 107 214 428 1391 2782
6232 and 6368 are an amicable pair with these proper divisors
6232 : 1 2 4 8 19 38 41 76 82 152 164 328 779 1558 3116
6368 : 1 2 4 8 16 32 199 398 796 1592 3184
10744 and 10856 are an amicable pair with these proper divisors
10744 : 1 2 4 8 17 34 68 79 136 158 316 632 1343 2686 5372
10856 : 1 2 4 8 23 46 59 92 118 184 236 472 1357 2714 5428
12285 and 14595 are an amicable pair with these proper divisors
12285 : 1 3 5 7 9 13 15 21 27 35 39 45 63 65 91 105 117 135 189 195 273 315 351 455 585 819 945 1365 1755 2457 4095
14595 : 1 3 5 7 15 21 35 105 139 417 695 973 2085 2919 4865
17296 and 18416 are an amicable pair with these proper divisors
17296 : 1 2 4 8 16 23 46 47 92 94 184 188 368 376 752 1081 2162 4324 8648
18416 : 1 2 4 8 16 1151 2302 4604 9208



## uBasic/4tH

Input "Limit: ";l Print "Amicable pairs < ";l

For n = 1 To l m = FUNC(_SumDivisors (n))-n If m = 0 Then Continue ' No division by zero, please p = FUNC(_SumDivisors (m))-m If (n=p) * (n<m) Then Print n;" and ";m Next

End

_LeastPower Param(2) Local(1)

c@ = a@ Do While (b@ % c@) = 0 c@ = c@ * a@ Loop

Return (c@)

' Return the sum of the proper divisors of a@

_SumDivisors Param(1) Local(4)

b@ = a@ c@ = 1

' Handle two specially

d@ = FUNC(_LeastPower (2,b@)) c@ = c@ * (d@ - 1) b@ = b@ / (d@ / 2)

' Handle odd factors

For e@ = 3 Step 2 While (e@*e@) < (b@+1) d@ = FUNC(_LeastPower (e@,b@)) c@ = c@ * ((d@ - 1) / (e@ - 1)) b@ = b@ / (d@ / e@) Loop

' At this point, t must be one or prime

If (b@ > 1) c@ = c@ * (b@+1) Return (c@)


{{Out}}

txt
Limit: 20000
Amicable pairs < 20000
220 and 284
1184 and 1210
2620 and 2924
5020 and 5564
6232 and 6368
10744 and 10856
12285 and 14595
17296 and 18416

0 OK, 0:238


## UTFool


···
http://rosettacode.org/wiki/Amicable_pairs
···
■ AmicablePairs
§ static
▶ main
• args⦂ String[]
∀ n ∈ 1…20000
m⦂ int: sumPropDivs n
if m < n = sumPropDivs m
System.out.println "⸨m⸩ ; ⸨n⸩"

▶ sumPropDivs⦂ int
• n⦂ int
m⦂ int: 1
∀ i ∈ √n ⋯> 1
m +: n \ i = 0 ? i + (i = n / i ? 0 ! n / i) ! 0
⏎ m



## VBA

Option Explicit

Public Sub AmicablePairs()
Dim a(2 To 20000) As Long, c As New Collection, i As Long, j As Long, t#
t = Timer
For i = LBound(a) To UBound(a)
'collect the sum of the proper divisors
'of each numbers between 2 and 20000
a(i) = S(i)
Next
'Double Loops to test the amicable
For i = LBound(a) To UBound(a)
For j = i + 1 To UBound(a)
If i = a(j) Then
If a(i) = j Then
On Error Resume Next
c.Add i & " : " & j, CStr(i * j)
On Error GoTo 0
Exit For
End If
End If
Next
Next
'End. Return :
Debug.Print "Execution Time : " & Timer - t & " seconds."
Debug.Print "Amicable pairs below 20 000 are : "
For i = 1 To c.Count
Debug.Print c.Item(i)
Next i
End Sub

Private Function S(n As Long) As Long
'returns the sum of the proper divisors of n
Dim j As Long
For j = 1 To n \ 2
If n Mod j = 0 Then S = j + S
Next
End Function


{{out}}

Execution Time : 7,95703125 seconds.
Amicable pairs below 20 000 are :
220 : 284
1184 : 1210
2620 : 2924
5020 : 5564
6232 : 6368
10744 : 10856
12285 : 14595
17296 : 18416


## VBScript

Not at all optimal. :-(

start = Now
Set nlookup = CreateObject("Scripting.Dictionary")
Set uniquepair = CreateObject("Scripting.Dictionary")

For i = 1 To 20000
sum = 0
For n = 1 To 20000
If n < i Then
If i Mod n = 0 Then
sum = sum + n
End If
End If
Next
Next

For j = 1 To 20000
sum = 0
For m = 1 To 20000
If m < j Then
If j Mod m = 0 Then
sum = sum + m
End If
End If
Next
If nlookup.Exists(sum) And nlookup.Item(sum) = j And j <> sum _
And uniquepair.Exists(sum) = False Then
End If
Next

For Each key In uniquepair.Keys
WScript.Echo key & ":" & uniquepair.Item(key)
Next

WScript.Echo "Execution Time: " & DateDiff("s",Start,Now) & " seconds"


{{out}}

220:284
1184:1210
2620:2924
5020:5564
6232:6368
10744:10856
12285:14595
17296:18416
Execution Time: 162 seconds


## Yabasic

{{trans|Lua}}

sub sumDivs(n)
local sum, d

sum = 1

for d = 2 to sqrt(n)
if not mod(n, d) then
sum = sum + d
sum = sum + n / d
end if
next
return sum
end sub

for n = 2 to 20000
m = sumDivs(n)
if m > n then
if sumDivs(m) = n print n, "\t", m
end if
next

print : print peek("millisrunning"), " ms"


## zkl

Slooooow

fcn properDivs(n){ [1.. (n + 1)/2 + 1].filter('wrap(x){ n%x==0 and n!=x }) }
const N=20000;
sums:=[1..N].pump(T(-1),fcn(n){ properDivs(n).sum(0) });
[0..].zip(sums).filter('wrap([(n,s)]){ (n<s<=N) and sums[s]==n }).println();


{{out}}


L(L(220,284),L(1184,1210),L(2620,2924),L(5020,5564),L(6232,6368),L(10744,10856),L(12285,14595),L(17296,18416))



## ZX Spectrum Basic

{{trans|AWK}}

10 LET limit=20000
20 PRINT "Amicable pairs < ";limit
30 FOR n=1 TO limit
40 LET num=n: GO SUB 1000
50 LET m=num
60 GO SUB 1000
70 IF n=num AND n<m THEN PRINT n;" ";m
80 NEXT n
90 STOP
1000 REM sumprop
1010 IF num<2 THEN LET num=0: RETURN
1020 LET sum=1
1030 LET root=SQR num
1040 FOR i=2 TO root-.01
1050 IF num/i=INT (num/i) THEN LET sum=sum+i+num/i
1060 NEXT i
1070 IF num/root=INT (num/root) THEN LET sum=sum+root
1080 LET num=sum
1090 RETURN


{{out}}

Amicable pairs < 20000
220 284
1184 1210
2620 2924
5020 5564
6232 6368
10744 10856
12285 14595
17296 18416