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{{task|Prime Numbers}} {{Wikipedia|Sexy_prime}}
In mathematics, '''sexy primes''' are prime numbers that differ from each other by six.
For example, the numbers '''5''' and '''11''' are both sexy primes, because '''11''' minus '''6''' is '''5'''.
The term "sexy prime" is a pun stemming from the Latin word for six: ''sex''.
'''Sexy prime pairs:''' Sexy prime pairs are groups of two primes that differ by '''6'''. e.g. '''(5 11), (7 13), (11 17)'''
See sequences: [[OEIS:A023201]] and [[OEIS:A046117]]
'''Sexy prime triplets:''' Sexy prime triplets are groups of three primes where each differs from the next by '''6'''. e.g. '''(5 11 17), (7 13 19), (17 23 29)'''
See sequences: [[OEIS:A046118]], [[OEIS:A046119]] and [[OEIS:A046120]]
'''Sexy prime quadruplets:''' Sexy prime quadruplets are groups of four primes where each differs from the next by '''6'''. e.g. '''(5 11 17 23), (11 17 23 29)'''
See sequences: [[OEIS:A023271]], [[OEIS:A046122]], [[OEIS:A046123]] and [[OEIS:A046124]]
'''Sexy prime quintuplets:''' Sexy prime quintuplets are groups of five primes with a common difference of '''6'''. One of the terms must be divisible by '''5''', because '''5''' and '''6''' are relatively prime. Thus, the only possible sexy prime quintuplet is '''(5 11 17 23 29)'''
;Task:
::*For each of pairs, triplets, quadruplets and quintuplets, Find and display the count of each group type of sexy primes less than one million thirty-five ('''1,000,035'''). ::*Display at most the '''last''' '''5''', less than one million thirty-five, of each sexy prime group type. ::*Find and display the count of the unsexy primes less than one million thirty-five. ::*Find and display the '''last 10''' unsexy primes less than one million thirty-five. ::*Note that 1000033 '''SHOULD NOT''' be counted in the pair count. It is sexy, but not in a pair within the limit. However, it also '''SHOULD NOT''' be listed in the unsexy primes since it is sexy.
AWK
# syntax: GAWK -f SEXY_PRIMES.AWK
BEGIN {
cutoff = 1000034
for (i=1; i<=cutoff; i++) {
n1 = i
if (is_prime(n1)) {
total_primes++
if ((n2 = n1 + 6) > cutoff) { continue }
if (is_prime(n2)) {
save(2,5,n1 FS n2)
if ((n3 = n2 + 6) > cutoff) { continue }
if (is_prime(n3)) {
save(3,5,n1 FS n2 FS n3)
if ((n4 = n3 + 6) > cutoff) { continue }
if (is_prime(n4)) {
save(4,5,n1 FS n2 FS n3 FS n4)
if ((n5 = n4 + 6) > cutoff) { continue }
if (is_prime(n5)) {
save(5,5,n1 FS n2 FS n3 FS n4 FS n5)
}
}
}
}
if ((s[2] s[3] s[4] s[5]) !~ (n1 "")) { # check for unsexy
save(1,10,n1)
}
}
}
printf("%d primes less than %s\n\n",total_primes,cutoff+1)
printf("%d unsexy primes\n%s\n\n",c[1],s[1])
printf("%d sexy prime pairs\n%s\n\n",c[2],s[2])
printf("%d sexy prime triplets\n%s\n\n",c[3],s[3])
printf("%d sexy prime quadruplets\n%s\n\n",c[4],s[4])
printf("%d sexy prime quintuplets\n%s\n\n",c[5],s[5])
exit(0)
}
function is_prime(x, i) {
if (x <= 1) {
return(0)
}
for (i=2; i<=int(sqrt(x)); i++) {
if (x % i == 0) {
return(0)
}
}
return(1)
}
function save(key,nbr_to_keep,str) {
c[key]++
str = s[key] str ", "
if (gsub(/,/,"&",str) > nbr_to_keep) {
str = substr(str,index(str,",")+2)
}
s[key] = str
}
{{out}}
78500 primes less than 1000035
48627 unsexy primes
999853, 999863, 999883, 999907, 999917, 999931, 999961, 999979, 999983, 1000003,
16386 sexy prime pairs
999371 999377, 999431 999437, 999721 999727, 999763 999769, 999953 999959,
2900 sexy prime triplets
997427 997433 997439, 997541 997547 997553, 998071 998077 998083, 998617 998623 998629, 998737 998743 998749,
325 sexy prime quadruplets
977351 977357 977363 977369, 983771 983777 983783 983789, 986131 986137 986143 986149, 990371 990377 990383 990389, 997091 997097 997103 997109,
1 sexy prime quintuplets
5 11 17 23 29,
C
Similar approach to the Go entry but only stores the arrays that need to be printed out.
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <locale.h>
#define TRUE 1
#define FALSE 0
typedef unsigned char bool;
void sieve(bool *c, int limit) {
int i, p = 3, p2;
// TRUE denotes composite, FALSE denotes prime.
c[0] = TRUE;
c[1] = TRUE;
// no need to bother with even numbers over 2 for this task
for (;;) {
p2 = p * p;
if (p2 >= limit) {
break;
}
for (i = p2; i < limit; i += 2*p) {
c[i] = TRUE;
}
for (;;) {
p += 2;
if (!c[p]) {
break;
}
}
}
}
void printHelper(const char *cat, int len, int lim, int n) {
const char *sp = strcmp(cat, "unsexy primes") ? "sexy prime " : "";
const char *verb = (len == 1) ? "is" : "are";
printf("Number of %s%s less than %'d = %'d\n", sp, cat, lim, len);
printf("The last %d %s:\n", n, verb);
}
void printArray(int *a, int len) {
int i;
printf("[");
for (i = 0; i < len; ++i) printf("%d ", a[i]);
printf("\b]");
}
int main() {
int i, ix, n, lim = 1000035;
int pairs = 0, trips = 0, quads = 0, quins = 0, unsexy = 2;
int pr = 0, tr = 0, qd = 0, qn = 0, un = 2;
int lpr = 5, ltr = 5, lqd = 5, lqn = 5, lun = 10;
int last_pr[5][2], last_tr[5][3], last_qd[5][4], last_qn[5][5];
int last_un[10];
bool *sv = calloc(lim - 1, sizeof(bool)); // all FALSE by default
setlocale(LC_NUMERIC, "");
sieve(sv, lim);
// get the counts first
for (i = 3; i < lim; i += 2) {
if (i > 5 && i < lim-6 && !sv[i] && sv[i-6] && sv[i+6]) {
unsexy++;
continue;
}
if (i < lim-6 && !sv[i] && !sv[i+6]) {
pairs++;
} else continue;
if (i < lim-12 && !sv[i+12]) {
trips++;
} else continue;
if (i < lim-18 && !sv[i+18]) {
quads++;
} else continue;
if (i < lim-24 && !sv[i+24]) {
quins++;
}
}
if (pairs < lpr) lpr = pairs;
if (trips < ltr) ltr = trips;
if (quads < lqd) lqd = quads;
if (quins < lqn) lqn = quins;
if (unsexy < lun) lun = unsexy;
// now get the last 'x' for each category
for (i = 3; i < lim; i += 2) {
if (i > 5 && i < lim-6 && !sv[i] && sv[i-6] && sv[i+6]) {
un++;
if (un > unsexy - lun) {
last_un[un + lun - 1 - unsexy] = i;
}
continue;
}
if (i < lim-6 && !sv[i] && !sv[i+6]) {
pr++;
if (pr > pairs - lpr) {
ix = pr + lpr - 1 - pairs;
last_pr[ix][0] = i; last_pr[ix][1] = i + 6;
}
} else continue;
if (i < lim-12 && !sv[i+12]) {
tr++;
if (tr > trips - ltr) {
ix = tr + ltr - 1 - trips;
last_tr[ix][0] = i; last_tr[ix][1] = i + 6;
last_tr[ix][2] = i + 12;
}
} else continue;
if (i < lim-18 && !sv[i+18]) {
qd++;
if (qd > quads - lqd) {
ix = qd + lqd - 1 - quads;
last_qd[ix][0] = i; last_qd[ix][1] = i + 6;
last_qd[ix][2] = i + 12; last_qd[ix][3] = i + 18;
}
} else continue;
if (i < lim-24 && !sv[i+24]) {
qn++;
if (qn > quins - lqn) {
ix = qn + lqn - 1 - quins;
last_qn[ix][0] = i; last_qn[ix][1] = i + 6;
last_qn[ix][2] = i + 12; last_qn[ix][3] = i + 18;
last_qn[ix][4] = i + 24;
}
}
}
printHelper("pairs", pairs, lim, lpr);
printf(" [");
for (i = 0; i < lpr; ++i) {
printArray(last_pr[i], 2);
printf("\b] ");
}
printf("\b]\n\n");
printHelper("triplets", trips, lim, ltr);
printf(" [");
for (i = 0; i < ltr; ++i) {
printArray(last_tr[i], 3);
printf("\b] ");
}
printf("\b]\n\n");
printHelper("quadruplets", quads, lim, lqd);
printf(" [");
for (i = 0; i < lqd; ++i) {
printArray(last_qd[i], 4);
printf("\b] ");
}
printf("\b]\n\n");
printHelper("quintuplets", quins, lim, lqn);
printf(" [");
for (i = 0; i < lqn; ++i) {
printArray(last_qn[i], 5);
printf("\b] ");
}
printf("\b]\n\n");
printHelper("unsexy primes", unsexy, lim, lun);
printf(" [");
printArray(last_un, lun);
printf("\b]\n");
free(sv);
return 0;
}
{{out}}
Number of sexy prime pairs less than 1,000,035 = 16,386
The last 5 are:
[[999371 999377] [999431 999437] [999721 999727] [999763 999769] [999953 999959]]
Number of sexy prime triplets less than 1,000,035 = 2,900
The last 5 are:
[[997427 997433 997439] [997541 997547 997553] [998071 998077 998083] [998617 998623 998629] [998737 998743 998749]]
Number of sexy prime quadruplets less than 1,000,035 = 325
The last 5 are:
[[977351 977357 977363 977369] [983771 983777 983783 983789] [986131 986137 986143 986149] [990371 990377 990383 990389] [997091 997097 997103 997109]]
Number of sexy prime quintuplets less than 1,000,035 = 1
The last 1 is:
[[5 11 17 23 29]]
Number of unsexy primes less than 1,000,035 = 48,627
The last 10 are:
[[999853 999863 999883 999907 999917 999931 999961 999979 999983 1000003]
=={{header|F_Sharp|F#}}== This task uses [http://www.rosettacode.org/wiki/Extensible_prime_generator#The_function Extensible Prime Generator (F#)]
// Sexy primes. Nigel Galloway: October 2nd., 2018
let n=pCache |> Seq.takeWhile(fun n->n<1000035) |> Seq.filter(fun n->(not (isPrime(n+6)) && (not isPrime(n-6))))) |> Array.ofSeq
printfn "There are %d unsexy primes less than 1,000,035. The last 10 are:" n.Length
Array.skip (n.Length-10) n |> Array.iter(fun n->printf "%d " n); printfn ""
let ni=pCache |> Seq.takeWhile(fun n->n<1000035) |> Seq.filter(fun n->isPrime(n-6)) |> Array.ofSeq
printfn "There are %d sexy prime pairs all components of which are less than 1,000,035. The last 5 are:" ni.Length
Array.skip (ni.Length-5) ni |> Array.iter(fun n->printf "(%d,%d) " (n-6) n); printfn ""
let nig=ni |> Array.filter(fun n->isPrime(n-12))
printfn "There are %d sexy prime triplets all components of which are less than 1,000,035. The last 5 are:" nig.Length
Array.skip (nig.Length-5) nig |> Array.iter(fun n->printf "(%d,%d,%d) " (n-12) (n-6) n); printfn ""
let nige=nig |> Array.filter(fun n->isPrime(n-18))
printfn "There are %d sexy prime quadruplets all components of which are less than 1,000,035. The last 5 are:" nige.Length
Array.skip (nige.Length-5) nige |> Array.iter(fun n->printf "(%d,%d,%d,%d) " (n-18) (n-12) (n-6) n); printfn ""
let nigel=nige |> Array.filter(fun n->isPrime(n-24))
printfn "There are %d sexy prime quintuplets all components of which are less than 1,000,035. The last 5 are:" nigel.Length
Array.skip (nigel.Length-5) nigel |> Array.iter(fun n->printf "(%d,%d,%d,%d,%d) " (n-24) (n-18) (n-12) (n-6) n); printfn ""
{{out}}
There are 48627 unsexy primes less than 1,000,035. The last 10 are:
999853 999863 999883 999907 999917 999931 999961 999979 999983 1000003
There are 16386 sexy prime pairs all components of which are less than 1,000,035. The last 5 are:
(999371,999377) (999431,999437) (999721,999727) (999763,999769) (999953,999959)
There are 2900 sexy prime triplets all components of which are less than 1,000,035. The last 5 are:
(997427,997433,997439) (997541,997547,997553) (998071,998077,998083) (998617,998623,998629) (998737,998743,998749)
There are 325 sexy prime quadruplets all components of which are less than 1,000,035. The last 5 are:
(977351,977357,977363,977369) (983771,983777,983783,983789) (986131.986137,986143,986149) (990371,990377,990383,990389) (997091,997097,997103,997109)
There are 1 sexy prime quintuplets all components of which are less than 1,000,035. The last 5 are:
(5,11,17,23,29)
Factor
USING: combinators.short-circuit fry interpolate io kernel
literals locals make math math.primes math.ranges prettyprint qw
sequences tools.memory.private ;
IN: rosetta-code.sexy-primes
CONSTANT: limit 1,000,035
CONSTANT: primes $[ limit primes-upto ]
CONSTANT: tuplet-names qw{ pair triplet quadruplet quintuplet }
: tuplet ( m n -- seq ) dupd 1 - 6 * + 6 <range> ;
: viable-tuplet? ( seq -- ? )
[ [ prime? ] [ limit < ] bi and ] all? ;
: sexy-tuplets ( n -- seq ) [ primes ] dip '[
[ _ tuplet dup viable-tuplet? [ , ] [ drop ] if ] each
] { } make ;
: ?last5 ( seq -- seq' ) 5 short tail* ;
: last5 ( seq -- str )
?last5 [ { } like unparse ] map " " join ;
:: tuplet-info ( n -- last5 l5-len num-tup limit tuplet-name )
n sexy-tuplets :> tup tup last5 tup ?last5 length tup length
commas limit commas n 2 - tuplet-names nth ;
: show-tuplets ( n -- )
tuplet-info
[I Number of sexy prime ${0}s < ${1}: ${2}I] nl
[I Last ${0}: ${1}I] nl nl ;
: unsexy-primes ( -- seq ) primes [
{ [ 6 + prime? not ] [ 6 - prime? not ] } 1&&
] filter ;
: show-unsexy ( -- )
unsexy-primes dup length commas limit commas
[I Number of unsexy primes < ${0}: ${1}I] nl
"Last 10: " write 10 short tail* [ pprint bl ] each nl ;
: main ( -- ) 2 5 [a,b] [ show-tuplets ] each show-unsexy ;
MAIN: main
{{out}}
Number of sexy prime pairs < 1,000,035: 16,386
Last 5: { 999371 999377 } { 999431 999437 } { 999721 999727 } { 999763 999769 } { 999953 999959 }
Number of sexy prime triplets < 1,000,035: 2,900
Last 5: { 997427 997433 997439 } { 997541 997547 997553 } { 998071 998077 998083 } { 998617 998623 998629 } { 998737 998743 998749 }
Number of sexy prime quadruplets < 1,000,035: 325
Last 5: { 977351 977357 977363 977369 } { 983771 983777 983783 983789 } { 986131 986137 986143 986149 } { 990371 990377 990383 990389 } { 997091 997097 997103 997109 }
Number of sexy prime quintuplets < 1,000,035: 1
Last 1: { 5 11 17 23 29 }
Number of unsexy primes < 1,000,035: 48,627
Last 10: 999853 999863 999883 999907 999917 999931 999961 999979 999983 1000003
Go
package main
import "fmt"
func sieve(limit int) []bool {
limit++
// True denotes composite, false denotes prime.
c := make([]bool, limit) // all false by default
c[0] = true
c[1] = true
// no need to bother with even numbers over 2 for this task
p := 3 // Start from 3.
for {
p2 := p * p
if p2 >= limit {
break
}
for i := p2; i < limit; i += 2 * p {
c[i] = true
}
for {
p += 2
if !c[p] {
break
}
}
}
return c
}
func commatize(n int) string {
s := fmt.Sprintf("%d", n)
if n < 0 {
s = s[1:]
}
le := len(s)
for i := le - 3; i >= 1; i -= 3 {
s = s[0:i] + "," + s[i:]
}
if n >= 0 {
return s
}
return "-" + s
}
func printHelper(cat string, le, lim, max int) (int, int, string) {
cle, clim := commatize(le), commatize(lim)
if cat != "unsexy primes" {
cat = "sexy prime " + cat
}
fmt.Printf("Number of %s less than %s = %s\n", cat, clim, cle)
last := max
if le < last {
last = le
}
verb := "are"
if last == 1 {
verb = "is"
}
return le, last, verb
}
func main() {
lim := 1000035
sv := sieve(lim - 1)
var pairs [][2]int
var trips [][3]int
var quads [][4]int
var quins [][5]int
var unsexy = []int{2, 3}
for i := 3; i < lim; i += 2 {
if i > 5 && i < lim-6 && !sv[i] && sv[i-6] && sv[i+6] {
unsexy = append(unsexy, i)
continue
}
if i < lim-6 && !sv[i] && !sv[i+6] {
pair := [2]int{i, i + 6}
pairs = append(pairs, pair)
} else {
continue
}
if i < lim-12 && !sv[i+12] {
trip := [3]int{i, i + 6, i + 12}
trips = append(trips, trip)
} else {
continue
}
if i < lim-18 && !sv[i+18] {
quad := [4]int{i, i + 6, i + 12, i + 18}
quads = append(quads, quad)
} else {
continue
}
if i < lim-24 && !sv[i+24] {
quin := [5]int{i, i + 6, i + 12, i + 18, i + 24}
quins = append(quins, quin)
}
}
le, n, verb := printHelper("pairs", len(pairs), lim, 5)
fmt.Printf("The last %d %s:\n %v\n\n", n, verb, pairs[le-n:])
le, n, verb = printHelper("triplets", len(trips), lim, 5)
fmt.Printf("The last %d %s:\n %v\n\n", n, verb, trips[le-n:])
le, n, verb = printHelper("quadruplets", len(quads), lim, 5)
fmt.Printf("The last %d %s:\n %v\n\n", n, verb, quads[le-n:])
le, n, verb = printHelper("quintuplets", len(quins), lim, 5)
fmt.Printf("The last %d %s:\n %v\n\n", n, verb, quins[le-n:])
le, n, verb = printHelper("unsexy primes", len(unsexy), lim, 10)
fmt.Printf("The last %d %s:\n %v\n\n", n, verb, unsexy[le-n:])
}
{{out}}
Number of sexy prime pairs less than 1,000,035 = 16,386
The last 5 are:
[[999371 999377] [999431 999437] [999721 999727] [999763 999769] [999953 999959]]
Number of sexy prime triplets less than 1,000,035 = 2,900
The last 5 are:
[[997427 997433 997439] [997541 997547 997553] [998071 998077 998083] [998617 998623 998629] [998737 998743 998749]]
Number of sexy prime quadruplets less than 1,000,035 = 325
The last 5 are:
[[977351 977357 977363 977369] [983771 983777 983783 983789] [986131 986137 986143 986149] [990371 990377 990383 990389] [997091 997097 997103 997109]]
Number of sexy prime quintuplets less than 1,000,035 = 1
The last 1 is:
[[5 11 17 23 29]]
Number of unsexy primes less than 1,000,035 = 48,627
The last 10 are:
[999853 999863 999883 999907 999917 999931 999961 999979 999983 1000003]
Julia
using Primes
function nextby6(n, a)
top = length(a)
i = n + 1
j = n + 2
k = n + 3
if n >= top
return n
end
possiblenext = a[n] + 6
if i <= top && possiblenext == a[i]
return i
elseif j <= top && possiblenext == a[j]
return j
elseif k <= top && possiblenext == a[k]
return k
end
return n
end
function lastones(dict, n)
arr = sort(collect(keys(dict)))
beginidx = max(1, length(arr) - n + 1)
arr[beginidx: end]
end
function lastoneslessthan(dict, n, ceiling)
arr = filter(y -> y < ceiling, lastones(dict, n+3))
beginidx = max(1, length(arr) - n + 1)
arr[beginidx: end]
end
function primesbysexiness(x)
twins = Dict{Int64, Array{Int64,1}}()
triplets = Dict{Int64, Array{Int64,1}}()
quadruplets = Dict{Int64, Array{Int64,1}}()
quintuplets = Dict{Int64, Array{Int64,1}}()
possibles = primes(x + 30)
singles = filter(y -> y <= x - 6, possibles)
unsexy = Dict(p => true for p in singles)
for (i, p) in enumerate(singles)
twinidx = nextby6(i, possibles)
if twinidx > i
delete!(unsexy, p)
delete!(unsexy, p + 6)
twins[p] = [i, twinidx]
tripidx = nextby6(twinidx, possibles)
if tripidx > twinidx
triplets[p] = [i, twinidx, tripidx]
quadidx = nextby6(tripidx, possibles)
if quadidx > tripidx
quadruplets[p] = [i, twinidx, tripidx, quadidx]
quintidx = nextby6(quadidx, possibles)
if quintidx > quadidx
quintuplets[p] = [i, twinidx, tripidx, quadidx, quintidx]
end
end
end
end
end
# Find and display the count of each group
println("There are:\n$(length(twins)) twins,\n",
"$(length(triplets)) triplets,\n",
"$(length(quadruplets)) quadruplets, and\n",
"$(length(quintuplets)) quintuplets less than $x.")
println("The last 5 twin primes start with ", lastoneslessthan(twins, 5, x - 6))
println("The last 5 triplet primes start with ", lastones(triplets, 5))
println("The last 5 quadruplet primes start with ", lastones(quadruplets, 5))
println("The quintuplet primes start with ", lastones(quintuplets, 5))
println("There are $(length(unsexy)) unsexy primes less than $x.")
lastunsexy = sort(collect(keys(unsexy)))[length(unsexy) - 9: end]
println("The last 10 unsexy primes are: $lastunsexy")
end
primesbysexiness(1000035)
{{output}}
There are:
16386 twins,
2900 triplets,
325 quadruplets, and
1 quintuplets less than 1000035.
The last 5 twin primes start with [999371, 999431, 999721, 999763, 999953]
The last 5 triplet primes start with [997427, 997541, 998071, 998617, 998737]
The last 5 quadruplet primes start with [977351, 983771, 986131, 990371, 997091]
The quintuplet primes start with [5]
There are 48627 unsexy primes less than 1000035.
The last 10 unsexy primes are: [999853, 999863, 999883, 999907, 999917, 999931, 999961, 999979, 999983, 1000003]
Kotlin
{{trans|Go}}
// Version 1.2.71
fun sieve(lim: Int): BooleanArray {
var limit = lim + 1
// True denotes composite, false denotes prime.
val c = BooleanArray(limit) // all false by default
c[0] = true
c[1] = true
// No need to bother with even numbers over 2 for this task.
var p = 3 // Start from 3.
while (true) {
val p2 = p * p
if (p2 >= limit) break
for (i in p2 until limit step 2 * p) c[i] = true
while (true) {
p += 2
if (!c[p]) break
}
}
return c
}
fun printHelper(cat: String, len: Int, lim: Int, max: Int): Pair<Int, String> {
val cat2 = if (cat != "unsexy primes") "sexy prime " + cat else cat
System.out.printf("Number of %s less than %d = %,d\n", cat2, lim, len)
val last = if (len < max) len else max
val verb = if (last == 1) "is" else "are"
return last to verb
}
fun main(args: Array<String>) {
val lim = 1_000_035
val sv = sieve(lim - 1)
val pairs = mutableListOf<List<Int>>()
val trips = mutableListOf<List<Int>>()
val quads = mutableListOf<List<Int>>()
val quins = mutableListOf<List<Int>>()
val unsexy = mutableListOf(2, 3)
for (i in 3 until lim step 2) {
if (i > 5 && i < lim - 6 && !sv[i] && sv[i - 6] && sv[i + 6]) {
unsexy.add(i)
continue
}
if (i < lim - 6 && !sv[i] && !sv[i + 6]) {
val pair = listOf(i, i + 6)
pairs.add(pair)
} else continue
if (i < lim - 12 && !sv[i + 12]) {
val trip = listOf(i, i + 6, i + 12)
trips.add(trip)
} else continue
if (i < lim - 18 && !sv[i + 18]) {
val quad = listOf(i, i + 6, i + 12, i + 18)
quads.add(quad)
} else continue
if (i < lim - 24 && !sv[i + 24]) {
val quin = listOf(i, i + 6, i + 12, i + 18, i + 24)
quins.add(quin)
}
}
var (n2, verb2) = printHelper("pairs", pairs.size, lim, 5)
System.out.printf("The last %d %s:\n %s\n\n", n2, verb2, pairs.takeLast(n2))
var (n3, verb3) = printHelper("triplets", trips.size, lim, 5)
System.out.printf("The last %d %s:\n %s\n\n", n3, verb3, trips.takeLast(n3))
var (n4, verb4) = printHelper("quadruplets", quads.size, lim, 5)
System.out.printf("The last %d %s:\n %s\n\n", n4, verb4, quads.takeLast(n4))
var (n5, verb5) = printHelper("quintuplets", quins.size, lim, 5)
System.out.printf("The last %d %s:\n %s\n\n", n5, verb5, quins.takeLast(n5))
var (nu, verbu) = printHelper("unsexy primes", unsexy.size, lim, 10)
System.out.printf("The last %d %s:\n %s\n\n", nu, verbu, unsexy.takeLast(nu))
}
{{output}}
Number of sexy prime pairs less than 1000035 = 16,386
The last 5 are:
[[999371, 999377], [999431, 999437], [999721, 999727], [999763, 999769], [999953, 999959]]
Number of sexy prime triplets less than 1000035 = 2,900
The last 5 are:
[[997427, 997433, 997439], [997541, 997547, 997553], [998071, 998077, 998083], [998617, 998623, 998629], [998737, 998743, 998749]]
Number of sexy prime quadruplets less than 1000035 = 325
The last 5 are:
[[977351, 977357, 977363, 977369], [983771, 983777, 983783, 983789], [986131, 986137, 986143, 986149], [990371, 990377, 990383, 990389], [997091, 997097, 997103, 997109]]
Number of sexy prime quintuplets less than 1000035 = 1
The last 1 is:
[[5, 11, 17, 23, 29]]
Number of unsexy primes less than 1000035 = 48,627
The last 10 are:
[999853, 999863, 999883, 999907, 999917, 999931, 999961, 999979, 999983, 1000003]
Pascal
{{works with|Free Pascal}} Is the count of unsexy primes = primes-2* SexyPrimesPairs +SexyPrimesTriplets-SexyPrimesQuintuplet?
48627 unsexy primes // = 78500-2*16386+2900-1
37907606 unsexy primes // = 50847538-2*6849047+758163-1 It seems so, not a proove.
program SexyPrimes;
uses
SysUtils;
const
ctext: array[0..5] of string = ('Primes',
'sexy prime pairs',
'sexy prime triplets',
'sexy prime quadruplets',
'sexy prime quintuplet',
'sexy prime sextuplet');
primeLmt = 1000 * 1000 + 35;
type
sxPrtpl = record
spCnt,
splast5Idx: nativeInt;
splast5: array[0..6] of NativeInt;
end;
var
sieve: array[0..primeLmt] of byte;
sexyPrimesTpl: array[0..5] of sxPrtpl;
unsexyprimes: NativeUint;
procedure dosieve;
var
p, delPos, fact: NativeInt;
begin
p := 2;
repeat
if sieve[p] = 0 then
begin
delPos := primeLmt div p;
if delPos < p then
BREAK;
fact := delPos * p;
while delPos >= p do
begin
if sieve[delPos] = 0 then
sieve[fact] := 1;
Dec(delPos);
Dec(fact, p);
end;
end;
Inc(p);
until False;
end;
procedure CheckforSexy;
var
i, idx, sieveMask, tstMask: NativeInt;
begin
sieveMask := -1;
for i := 2 to primelmt do
begin
tstMask := 1;
sieveMask := sieveMask + sieveMask + sieve[i];
idx := 0;
repeat
if (tstMask and sieveMask) = 0 then
with sexyPrimesTpl[idx] do
begin
Inc(spCnt);
//memorize the last entry
Inc(splast5idx);
if splast5idx > 5 then
splast5idx := 1;
splast5[splast5idx] := i;
tstMask := tstMask shl 6 + 1;
end
else
begin
BREAK;
end;
Inc(idx);
until idx > 5;
end;
end;
procedure CheckforUnsexy;
var
i: NativeInt;
begin
for i := 2 to 6 do
begin
if (Sieve[i] = 0) and (Sieve[i + 6] = 1) then
Inc(unsexyprimes);
end;
for i := 2 + 6 to primelmt - 6 do
begin
if (Sieve[i] = 0) and (Sieve[i - 6] = 1) and (Sieve[i + 6] = 1) then
Inc(unsexyprimes);
end;
end;
procedure OutLast5(idx: NativeInt);
var
i, j, k: nativeInt;
begin
with sexyPrimesTpl[idx] do
begin
writeln(cText[idx], ' ', spCnt);
i := splast5idx + 1;
for j := 1 to 5 do
begin
if i > 5 then
i := 1;
if splast5[i] <> 0 then
begin
Write('[');
for k := idx downto 1 do
Write(splast5[i] - k * 6, ' ');
Write(splast5[i], ']');
end;
Inc(i);
end;
end;
writeln;
end;
procedure OutLastUnsexy(cnt:NativeInt);
var
i: NativeInt;
erg: array of NativeUint;
begin
if cnt < 1 then
EXIT;
setlength(erg,cnt);
dec(cnt);
if cnt < 0 then
EXIT;
for i := primelmt downto 2 + 6 do
begin
if (Sieve[i] = 0) and (Sieve[i - 6] = 1) and (Sieve[i + 6] = 1) then
Begin
erg[cnt] := i;
dec(cnt);
If cnt < 0 then
BREAK;
end;
end;
write('the last ',High(Erg)+1,' unsexy primes ');
For i := 0 to High(erg)-1 do
write(erg[i],',');
write(erg[High(erg)]);
end;
var
T1, T0: int64;
i: nativeInt;
begin
T0 := GettickCount64;
dosieve;
T1 := GettickCount64;
writeln('Sieving is done in ', T1 - T0, ' ms');
T0 := GettickCount64;
CheckforSexy;
T1 := GettickCount64;
writeln('Checking is done in ', T1 - T0, ' ms');
unsexyprimes := 0;
T0 := GettickCount64;
CheckforUnsexy;
T1 := GettickCount64;
writeln('Checking unsexy is done in ', T1 - T0, ' ms');
writeln('Limit : ', primelmt);
for i := 0 to 4 do
begin
OutLast5(i);
end;
writeln;
writeln(unsexyprimes,' unsexy primes');
OutLastUnsexy(10);
end.
{{Output}}
Sieving is done in 361 ms
Checking is done in 2 ms
Checking unsexy is done in 1 ms
Limit : 1000035
Primes 78500
[999961][999979][999983][1000003][1000033]
sexy prime pairs 16386
[999371 999377][999431 999437][999721 999727][999763 999769][999953 999959]
sexy prime triplets 2900
[997427 997433 997439][997541 997547 997553][998071 998077 998083][998617 998623 998629][998737 998743 998749]
sexy prime quadruplets 325
[977351 977357 977363 977369][983771 983777 983783 983789][986131 986137 986143 986149][990371 990377 990383 990389][997091 997097 997103 997109]
sexy prime quintuplet 1
[5 11 17 23 29]
48627 unsexy primes
the last 10 unsexy primes 999853,999863,999883,999907,999917,999931,999961,999979,999983,1000003
---
Sieving is done in 5248 ms
Checking is done in 1462 ms
Checking unsexy is done in 1062 ms
Limit : 1000000035
Primes 50847538
[999999937][1000000007][1000000009][1000000021][1000000033]
sexy prime pairs 6849047
[999999191 999999197][999999223 999999229][999999607 999999613][999999733 999999739][999999751 999999757]
sexy prime triplets 758163
[999990347 999990353 999990359][999993811 999993817 999993823][999994427 999994433 999994439][999994741 999994747 999994753][999996031 999996037 999996043]
sexy prime quadruplets 56643
[999835261 999835267 999835273 999835279][999864611 999864617 999864623 999864629][999874021 999874027 999874033 999874039][999890981 999890987 999890993 999890999][999956921 999956927 999956933 999956939]
sexy prime quintuplet 1
[5 11 17 23 29]
37907606 unsexy primes // = 50847538-2*6849047+758163-1
the last 10 unsexy primes 999999677,999999761,999999797,999999883,999999893,999999929,999999937,1000000007,1000000009,1000000021
Perl
{{libheader|ntheory}}
We will use the prime iterator and primality test from the ntheory module.
use ntheory qw/prime_iterator is_prime/;
sub tuple_tail {
my($n,$cnt,@array) = @_;
$n = @array if $n > @array;
my @tail;
for (1..$n) {
my $p = $array[-$n+$_-1];
push @tail, "(" . join(" ", map { $p+6*$_ } 0..$cnt-1) . ")";
}
return @tail;
}
sub comma {
(my $s = reverse shift) =~ s/(.{3})/$1,/g;
($s = reverse $s) =~ s/^,//;
return $s;
}
sub sexy_string { my $p = shift; is_prime($p+6) || is_prime($p-6) ? 'sexy' : 'unsexy' }
my $max = 1_000_035;
my $cmax = comma $max;
my $iter = prime_iterator;
my $p = $iter->();
my %primes;
push @{$primes{sexy_string($p)}}, $p;
while ( ($p = $iter->()) < $max) {
push @{$primes{sexy_string($p)}}, $p;
$p+ 6 < $max && is_prime($p+ 6) ? push @{$primes{'pair'}}, $p : next;
$p+12 < $max && is_prime($p+12) ? push @{$primes{'triplet'}}, $p : next;
$p+18 < $max && is_prime($p+18) ? push @{$primes{'quadruplet'}}, $p : next;
$p+24 < $max && is_prime($p+24) ? push @{$primes{'quintuplet'}}, $p : next;
}
print "Total primes less than $cmax: " . comma(@{$primes{'sexy'}} + @{$primes{'unsexy'}}) . "\n\n";
for (['pair', 2], ['triplet', 3], ['quadruplet', 4], ['quintuplet', 5]) {
my($sexy,$cnt) = @$_;
print "Number of sexy prime ${sexy}s less than $cmax: " . comma(scalar @{$primes{$sexy}}) . "\n";
print " Last 5 sexy prime ${sexy}s less than $cmax: " . join(' ', tuple_tail(5,$cnt,@{$primes{$sexy}})) . "\n";
print "\n";
}
print "Number of unsexy primes less than $cmax: ". comma(scalar @{$primes{unsexy}}) . "\n";
print " Last 10 unsexy primes less than $cmax: ". join(' ', @{$primes{unsexy}}[-10..-1]) . "\n";
{{out}}
Total primes less than 1,000,035: 78,500
Number of sexy prime pairs less than 1,000,035: 16,386
Last 5 sexy prime pairs less than 1,000,035: (999371 999377) (999431 999437) (999721 999727) (999763 999769) (999953 999959)
Number of sexy prime triplets less than 1,000,035: 2,900
Last 5 sexy prime triplets less than 1,000,035: (997427 997433 997439) (997541 997547 997553) (998071 998077 998083) (998617 998623 998629) (998737 998743 998749)
Number of sexy prime quadruplets less than 1,000,035: 325
Last 5 sexy prime quadruplets less than 1,000,035: (977351 977357 977363 977369) (983771 983777 983783 983789) (986131 986137 986143 986149) (990371 990377 990383 990389) (997091 997097 997103 997109)
Number of sexy prime quintuplets less than 1,000,035: 1
Last 5 sexy prime quintuplets less than 1,000,035: (5 11 17 23 29)
Number of unsexy primes less than 1,000,035: 48,627
Last 10 unsexy primes less than 1,000,035: 999853 999863 999883 999907 999917 999931 999961 999979 999983 1000003
Using cluster sieve
The ntheory module includes a function to do very efficient sieving for prime clusters. Even though we are doing repeated work for this task, it is still faster than the previous code. The helper subroutines and output code remain identical, as does the generated output.
The cluster sieve becomes more efficient as the number of terms increases. See for example [[oeis:a213646|OEIS Prime 11-tuplets]].
use ntheory qw/sieve_prime_cluster forprimes is_prime/;
# ... identical helper functions
my %primes = (
sexy => [],
unsexy => [],
pair => [ sieve_prime_cluster(1, $max-1- 6, 6) ],
triplet => [ sieve_prime_cluster(1, $max-1-12, 6, 12) ],
quadruplet => [ sieve_prime_cluster(1, $max-1-18, 6, 12, 18) ],
quintuplet => [ sieve_prime_cluster(1, $max-1-24, 6, 12, 18, 24) ],
);
forprimes {
push @{$primes{sexy_string($_)}}, $_;
} $max-1;
# ... identical output code
Perl 6
{{works with|Rakudo|2018.08}}
use Math::Primesieve;
my $sieve = Math::Primesieve.new;
my $max = 1_000_035;
my @primes = $sieve.primes($max);
my $filter = @primes.Set;
my $primes = @primes.categorize: &sexy;
say "Total primes less than {comma $max}: ", comma +@primes;
for <pair 2 triplet 3 quadruplet 4 quintuplet 5> -> $sexy, $cnt {
say "Number of sexy prime {$sexy}s less than {comma $max}: ", comma +$primes{$sexy};
say " Last 5 sexy prime {$sexy}s less than {comma $max}: ",
join ' ', $primes{$sexy}.tail(5).grep(*.defined).map:
{ "({ $_ «+« (0,6 … 24)[^$cnt] })" }
say '';
}
say "Number of unsexy primes less than {comma $max}: ", comma +$primes<unsexy>;
say " Last 10 unsexy primes less than {comma $max}: ", $primes<unsexy>.tail(10);
sub sexy ($i) {
gather {
take 'quintuplet' if all($filter{$i «+« (6,12,18,24)});
take 'quadruplet' if all($filter{$i «+« (6,12,18)});
take 'triplet' if all($filter{$i «+« (6,12)});
take 'pair' if $filter{$i + 6};
take (($i >= $max - 6) && ($i + 6).is-prime) ||
(so any($filter{$i «+« (6, -6)})) ?? 'sexy' !! 'unsexy';
}
}
sub comma { $^i.flip.comb(3).join(',').flip }
{{out}}
Total primes less than 1,000,035: 78,500
Number of sexy prime pairs less than 1,000,035: 16,386
Last 5 sexy prime pairs less than 1,000,035: (999371 999377) (999431 999437) (999721 999727) (999763 999769) (999953 999959)
Number of sexy prime triplets less than 1,000,035: 2,900
Last 5 sexy prime triplets less than 1,000,035: (997427 997433 997439) (997541 997547 997553) (998071 998077 998083) (998617 998623 998629) (998737 998743 998749)
Number of sexy prime quadruplets less than 1,000,035: 325
Last 5 sexy prime quadruplets less than 1,000,035: (977351 977357 977363 977369) (983771 983777 983783 983789) (986131 986137 986143 986149) (990371 990377 990383 990389) (997091 997097 997103 997109)
Number of sexy prime quintuplets less than 1,000,035: 1
Last 5 sexy prime quintuplets less than 1,000,035: (5 11 17 23 29)
Number of unsexy primes less than 1,000,035: 48,627
Last 10 unsexy primes less than 1,000,035: (999853 999863 999883 999907 999917 999931 999961 999979 999983 1000003)
Phix
function create_sieve(integer limit)
sequence sieve = repeat(true,limit)
sieve[1] = false
for i=4 to limit by 2 do
sieve[i] = false
end for
for p=3 to floor(sqrt(limit)) by 2 do
integer p2 = p*p
if sieve[p2] then
for k=p2 to limit by p*2 do
sieve[k] = false
end for
end if
end for
return sieve
end function
constant lim = 1000035,
--constant lim = 100, -- (this works too)
limit = lim-(and_bits(lim,1)=0), -- (limit must be odd)
sieve = create_sieve(limit+6) -- (+6 to check for sexiness)
sequence sets = repeat({},5), -- (unsexy,pairs,trips,quads,quins)
limits = {10,5,4,3,1},
counts = 1&repeat(0,4) -- (2 is an unsexy prime)
integer total = 1 -- ""
for i=limit to 3 by -2 do -- (this loop skips 2)
if sieve[i] then
total += 1
if sieve[i+6]=false and (i-6<0 or sieve[i-6]=false) then
counts[1] += 1 -- unsexy
if length(sets[1])<limits[1] then
sets[1] = prepend(sets[1],i)
end if
else
sequence set = {i}
for j=i-6 to 3 by -6 do
if j<=0 or sieve[j]=false then exit end if
set = prepend(set,j)
integer l = length(set)
if length(sets[l])<limits[l] then
sets[l] = prepend(sets[l],set)
end if
counts[l] += 1
end for
end if
end if
end for
if length(sets[1])<limits[1] then
sets[1] = prepend(sets[1],2) -- (as 2 skipped above)
end if
constant fmt = """
Of %,d primes less than %,d there are:
%,d unsexy primes, the last %d being %s
%,d pairs, the last %d being %s
%,d triplets, the last %d being %s
%,d quadruplets, the last %d being %s
%,d quintuplet, the last %d being %s
"""
sequence results = {total,lim,
0,0,"",
0,0,"",
0,0,"",
0,0,"",
0,0,""}
for i=1 to 5 do
results[i*3..i*3+2] = {counts[i],length(sets[i]),sprint(sets[i])}
end for
printf(1,fmt,results)
{{out}}
Of 78,500 primes less than 1,000,035 there are:
48,627 unsexy primes, the last 10 being {999853,999863,999883,999907,999917,999931,999961,999979,999983,1000003}
16,386 pairs, the last 5 being {{999371,999377},{999431,999437},{999721,999727},{999763,999769},{999953,999959}}
2,900 triplets, the last 4 being {{997541,997547,997553},{998071,998077,998083},{998617,998623,998629},{998737,998743,998749}}
325 quadruplets, the last 3 being {{986131,986137,986143,986149},{990371,990377,990383,990389},{997091,997097,997103,997109}}
1 quintuplet, the last 1 being {{5,11,17,23,29}}
Python
Imperative Style
LIMIT = 1_000_035
def primes2(limit=LIMIT):
if limit < 2: return []
if limit < 3: return [2]
lmtbf = (limit - 3) // 2
buf = [True] * (lmtbf + 1)
for i in range((int(limit ** 0.5) - 3) // 2 + 1):
if buf[i]:
p = i + i + 3
s = p * (i + 1) + i
buf[s::p] = [False] * ((lmtbf - s) // p + 1)
return [2] + [i + i + 3 for i, v in enumerate(buf) if v]
primes = primes2(LIMIT +6)
primeset = set(primes)
primearray = [n in primeset for n in range(LIMIT)]
#%%
s = [[] for x in range(4)]
unsexy = []
for p in primes:
if p > LIMIT:
break
if p + 6 in primeset and p + 6 < LIMIT:
s[0].append((p, p+6))
elif p + 6 in primeset:
break
else:
if p - 6 not in primeset:
unsexy.append(p)
continue
if p + 12 in primeset and p + 12 < LIMIT:
s[1].append((p, p+6, p+12))
else:
continue
if p + 18 in primeset and p + 18 < LIMIT:
s[2].append((p, p+6, p+12, p+18))
else:
continue
if p + 24 in primeset and p + 24 < LIMIT:
s[3].append((p, p+6, p+12, p+18, p+24))
#%%
print('"SEXY" PRIME GROUPINGS:')
for sexy, name in zip(s, 'pairs triplets quadruplets quintuplets'.split()):
print(f' {len(sexy)} {na (not isPrime(n-6))))) |> Array.ofSeq
printfn "There are %d unsexy primes less than 1,000,035. The last 10 are:" n.Length
Array.skip (n.Length-10) n |> Array.iter(fun n->printf "%d " n); printfn ""
let ni=pCache |> Seq.takeWhile(fun n->nme} ending with ...')
for sx in sexy[-5:]:
print(' ',sx)
print(f'\nThere are {len(unsexy)} unsexy primes ending with ...')
for usx in unsexy[-10:]:
print(' ',usx)
{{out}}
"SEXY" PRIME GROUPINGS:
16386 pairs ending with ...
(999371, 999377)
(999431, 999437)
(999721, 999727)
(999763, 999769)
(999953, 999959)
2900 triplets ending with ...
(997427, 997433, 997439)
(997541, 997547, 997553)
(998071, 998077, 998083)
(998617, 998623, 998629)
(998737, 998743, 998749)
325 quadruplets ending with ...
(977351, 977357, 977363, 977369)
(983771, 983777, 983783, 983789)
(986131, 986137, 986143, 986149)
(990371, 990377, 990383, 990389)
(997091, 997097, 997103, 997109)
1 quintuplets ending with ...
(5, 11, 17, 23, 29)
There are 48627 unsexy primes ending with ...
999853
999863
999883
999907
999917
999931
999961
999979
999983
1000003
Functional style
{{trans|FSharp}} This task uses [[Extensible_prime_generator#210-wheel_postponed_incremental_sieve]]
#Functional Sexy Primes. Nigel Galloway: October 5th., 2018
from itertools import *
z=primes()
n=frozenset(takewhile(lambda x: x<1000035,z))
ni=sorted(list(filter(lambda g: n.__contains__(g+6) ,n)))
print ("There are",len(ni),"sexy prime pairs all components of which are less than 1,000,035. The last 5 are:")
for g in islice(ni,max(len(ni)-5,0),len(ni)): print(format("(%d,%d) " % (g,g+6)))
nig=list(filter(lambda g: n.__contains__(g+12) ,ni))
print ("There are",len(nig),"sexy prime triplets all components of which are less than 1,000,035. The last 5 are:")
for g in islice(nig,max(len(nig)-5,0),len(nig)): print(format("(%d,%d,%d) " % (g,g+6,g+12)))
nige=list(filter(lambda g: n.__contains__(g+18) ,nig))
print ("There are",len(nige),"sexy prime quadruplets all components of which are less than 1,000,035. The last 5 are:")
for g in islice(nige,max(len(nige)-5,0),len(nige)): print(format("(%d,%d,%d,%d) " % (g,g+6,g+12,g+18)))
nigel=list(filter(lambda g: n.__contains__(g+24) ,nige))
print ("There are",len(nigel),"sexy prime quintuplets all components of which are less than 1,000,035. The last 5 are:")
for g in islice(nigel,max(len(nigel)-5,0),len(nigel)): print(format("(%d,%d,%d,%d,%d) " % (g,g+6,g+12,g+18,g+24)))
un=frozenset(takewhile(lambda x: x<1000050,z)).union(n)
unsexy=sorted(list(filter(lambda g: not un.__contains__(g+6) and not un.__contains__(g-6),n)))
print ("There are",len(unsexy),"unsexy primes less than 1,000,035. The last 10 are:")
for g in islice(unsexy,max(len(unsexy)-10,0),len(unsexy)): print(g)
{{out}}
There are 16386 sexy prime pairs all components of which are less than 1,000,035. The last 5 are:
(999371,999377)
(999431,999437)
(999721,999727)
(999763,999769)
(999953,999959)
There are 2900 sexy prime triplets all components of which are less than 1,000,035. The last 5 are:
(997427,997433,997439)
(997541,997547,997553)
(998071,998077,998083)
(998617,998623,998629)
(998737,998743,998749)
There are 325 sexy prime quadruplets all components of which are less than 1,000,035. The last 5 are:
(977351,977357,977363,977369)
(983771,983777,983783,983789)
(986131,986137,986143,986149)
(990371,990377,990383,990389)
(997091,997097,997103,997109)
There are 1 sexy prime quintuplets all components of which are less than 1,000,035. The last 5 are:
(5,11,17,23,29)
There are 48627 unsexy primes less than 1,000,035. The last 10 are:
999853
999863
999883
999907
999917
999931
999961
999979
999983
1000003
REXX
/*REXX program finds and displays various kinds of sexy and unsexy primes less than N.*/
parse arg N endU end2 end3 end4 end5 . /*obtain optional argument from the CL.*/
if N=='' | N=="," then N= 1000035 - 1 /*Not specified? Then use the default.*/
if endU=='' | endU=="," then endU= 10 /* " " " " " " */
if end2=='' | end2=="," then end2= 5 /* " " " " " " */
if end3=='' | end3=="," then end3= 5 /* " " " " " " */
if end4=='' | end4=="," then end4= 5 /* " " " " " " */
if end5=='' | end5=="," then end4= 5 /* " " " " " " */
call genSq /*gen some squares for the DO k=7 UNTIL*/
call genPx /* " prime (@.) & sexy prime (X.) array*/
call genXU /*gen lists, types of sexy Ps, unsexy P*/
call getXs /*gen lists, last # of types of sexy Ps*/
@sexy= ' sexy prime' /*a handy literal for some of the SAYs.*/
w2= words( translate(x2,, '~') ); y2= words(x2) /*count #primes in the sexy pairs. */
w3= words( translate(x3,, '~') ); y3= words(x3) /* " " " " " " triplets. */
w4= words( translate(x4,, '~') ); y4= words(x4) /* " " " " " " quadruplets*/
w5= words( translate(x5,, '~') ); y5= words(x5) /* " " " " " " quintuplets*/
say 'There are ' commas(w2%2) @sexy "pairs less than " Nc
say 'The last ' commas(end2) @sexy "pairs are:"; say subword(x2, max(1,y2-end2+1))
say
say 'There are ' commas(w3%3) @sexy "triplets less than " Nc
say 'The last ' commas(end3) @sexy "triplets are:"; say subword(x3, max(1,y3-end3+1))
say
say 'There are ' commas(w4%4) @sexy "quadruplets less than " Nc
say 'The last ' commas(end4) @sexy "quadruplets are:"; say subword(x4, max(1,y4-end4+1))
say
say 'There is ' commas(w5%5) @sexy "quintuplet less than " Nc
say 'The last ' commas(end4) @sexy "quintuplet are:"; say subword(x5, max(1,y5-end4+1))
say
say 'There are ' commas(s1) " sexy primes less than " Nc
say 'There are ' commas(u1) " unsexy primes less than " Nc
say 'The last ' commas(endU) " unsexy primes are: " subword(u, max(1,u1-endU+1))
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
commas: procedure; parse arg _; n= _'.9'; #= 123456789; b= verify(n, #, "M")
e= verify(n, #'0', , verify(n, #"0.", 'M') ) - 4
do j=e to b by -3; _= insert(',', _, j); end /*j*/; return _
/*──────────────────────────────────────────────────────────────────────────────────────*/
genSQ: do i=17 by 2 until i**2 > N+7; s.i= i**2; end; return /*S used for square roots*/
/*──────────────────────────────────────────────────────────────────────────────────────*/
genPx: @.=; #= 0; !.= 0. /*P array; P count; sexy P array*/
if N>1 then do; #= 1; @.1= 2; !.2= 1; end /*count of primes found (so far)*/
x.=!.; LPs=3 5 7 11 13 17 /*sexy prime array; low P list.*/
do j=3 by 2 to N+6 /*start in the cellar & work up.*/
if j<19 then if wordpos(j, LPs)==0 then iterate
else do; #= #+1; @.#= j; !.j= 1; b= j - 6
if !.b then x.b= 1; iterate
end
if j// 3 ==0 then iterate /* ··· and eliminate multiples of 3.*/
parse var j '' -1 _ /* get the rightmost digit of J. */
if _ ==5 then iterate /* ··· and eliminate multiples of 5.*/
if j// 7 ==0 then iterate /* ··· " " " " 7.*/
if j//11 ==0 then iterate /* ··· " " " " 11.*/
if j//13 ==0 then iterate /* ··· " " " " 13.*/
do k=7 until s._ > j; _= @.k /*÷ by primes starting at 7th prime. */
if j // _ == 0 then iterate j /*get the remainder of j÷@.k ___ */
end /*k*/ /*divide up through & including √ J */
if j<=N then do; #= #+1; @.#= j; end /*bump P counter; assign prime to @.*/
!.j= 1 /*define Jth number as being prime.*/
b= j - 6 /*B: lower part of a sexy prime pair?*/
if !.b then do; x.b=1; if j<=N then x.j=1; end /*assign (both parts ?) sexy Ps.*/
end /*j*/; return
/*──────────────────────────────────────────────────────────────────────────────────────*/
genXU: u= 2; Nc=commas(N+1); s= /*1st unsexy prime; add commas to N+1*/
say 'There are ' commas(#) " primes less than " Nc; say
do k=2 for #-1; p= @.k; if x.p then s=s p /*if sexy prime, add it to list*/
else u= u p /* " unsexy " " " " " */
end /*k*/ /* [↑] traispe through odd Ps. */
s1= words(s); u1= words(u); return /*# of sexy primes; # unsexy primes.*/
/*──────────────────────────────────────────────────────────────────────────────────────*/
getXs: x2=; do k=2 for #-1; p=@.k; if \x.p then iterate /*build sexy prime list. */
b=p- 6; if \x.b then iterate; x2=x2 b'~'p
end /*k*/
x3=; do k=2 for #-1; p=@.k; if \x.p then iterate /*build sexy P triplets. */
b=p- 6; if \x.b then iterate
t=p-12; if \x.t then iterate; x3=x3 t'~' || b"~"p
end /*k*/
x4=; do k=2 for #-1; p=@.k; if \x.p then iterate /*build sexy P quads. */
b=p- 6; if \x.b then iterate
t=p-12; if \x.t then iterate
q=p-18; if \x.q then iterate; x4=x4 q'~'t"~" || b'~'p
end /*k*/
x5=; do k=2 for #-1; p=@.k; if \x.p then iterate /*build sexy P quints. */
b=p- 6; if \x.b then iterate
t=p-12; if \x.t then iterate
q=p-18; if \x.q then iterate
v=p-24; if \x.v then iterate; x5=x5 v'~'q"~"t'~' || b"~"p
end /*k*/; return
{{out|output|text= when using the default inputs:}}
(Shown at '''5/6''' size.)
There are 78,500 primes less than 1,000,035
There are 16,386 sexy prime pairs less than 1,000,035
The last 5 sexy prime pairs are:
999371~999377 999431~999437 999721~999727 999763~999769 999953~999959
There are 2,900 sexy prime triplets less than 1,000,035
The last 5 sexy prime triplets are:
997427~997433~997439 997541~997547~997553 998071~998077~998083 998617~998623~998629 998737~998743~998749
There are 325 sexy prime quadruplets less than 1,000,035
The last 5 sexy prime quadruplets are:
977351~977357~977363~977369 983771~983777~983783~983789 986131~986137~986143~986149 990371~990377~990383~990389 997091~997097~997103~997109
There is 1 sexy prime quintuplet less than 1,000,035
The last 5 sexy prime quintuplet are:
5~11~17~23~29
There are 29,873 sexy primes less than 1,000,035
There are 48,627 unsexy primes less than 1,000,035
The last 10 unsexy primes are: 999853 999863 999883 999907 999917 999931 999961 999979 999983 1000003
```
## Ruby
```Ruby
require 'prime'
prime_array, sppair2, sppair3, sppair4, sppair5 = Array.new(5) {Array.new()} # arrays for prime numbers and index number to array for each pair.
unsexy, i, start = [2], 0, Time.now
Prime.each(1_000_100) {|prime| prime_array.push prime}
while prime_array[i] < 1_000_035
i+=1
unsexy.push(i) if prime_array[(i+1)..(i+2)].include?(prime_array[i]+6) == false && prime_array[(i-2)..(i-1)].include?(prime_array[i]-6) == false && prime_array[i]+6 < 1_000_035
prime_array[(i+1)..(i+4)].include?(prime_array[i]+6) && prime_array[i]+6 < 1_000_035 ? sppair2.push(i) : next
prime_array[(i+2)..(i+5)].include?(prime_array[i]+12) && prime_array[i]+12 < 1_000_035 ? sppair3.push(i) : next
prime_array[(i+3)..(i+6)].include?(prime_array[i]+18) && prime_array[i]+18 < 1_000_035 ? sppair4.push(i) : next
prime_array[(i+4)..(i+7)].include?(prime_array[i]+24) && prime_array[i]+24 < 1_000_035 ? sppair5.push(i) : next
end
puts "\nSexy prime pairs: #{sppair2.size} found:"
sppair2.last(5).each {|prime| print [prime_array[prime], prime_array[prime]+6].join(" - "), "\n"}
puts "\nSexy prime triplets: #{sppair3.size} found:"
sppair3.last(5).each {|prime| print [prime_array[prime], prime_array[prime]+6, prime_array[prime]+12].join(" - "), "\n"}
puts "\nSexy prime quadruplets: #{sppair4.size} found:"
sppair4.last(5).each {|prime| print [prime_array[prime], prime_array[prime]+6, prime_array[prime]+12, prime_array[prime]+18].join(" - "), "\n"}
puts "\nSexy prime quintuplets: #{sppair5.size} found:"
sppair5.last(5).each {|prime| print [prime_array[prime], prime_array[prime]+6, prime_array[prime]+12, prime_array[prime]+18, prime_array[prime]+24].join(" - "), "\n"}
puts "\nUnSexy prime: #{unsexy.size} found. Last 10 are:"
unsexy.last(10).each {|item| print prime_array[item], " "}
print "\n\n", Time.now - start, " seconds"
```
Output:
```txt
ruby 2.5.3p105 (2018-10-18 revision 65156) [x64-mingw32]
Sexy prime pairs: 16386 found:
999371 - 999377
999431 - 999437
999721 - 999727
999763 - 999769
999953 - 999959
Sexy prime triplets: 2900 found:
997427 - 997433 - 997439
997541 - 997547 - 997553
998071 - 998077 - 998083
998617 - 998623 - 998629
998737 - 998743 - 998749
Sexy prime quadruplets: 325 found:
977351 - 977357 - 977363 - 977369
983771 - 983777 - 983783 - 983789
986131 - 986137 - 986143 - 986149
990371 - 990377 - 990383 - 990389
997091 - 997097 - 997103 - 997109
Sexy prime quintuplets: 1 found:
5 - 11 - 17 - 23 - 29
UnSexy prime: 48627 found. Last 10 are:
999853 999863 999883 999907 999917 999931 999961 999979 999983 1000003
0.176955 seconds
```
## Sidef
```ruby
var limit = 1e6+35
var primes = limit.primes
say "Total number of primes <= #{limit.commify} is #{primes.len.commify}."
say "Sexy k-tuple primes <= #{limit.commify}:\n"
(2..5).each {|k|
var groups = []
primes.each {|p|
var group = (1..^k -> map {|j| 6*j + p })
if (group.all{.is_prime} && (group[-1] <= limit)) {
groups << [p, group...]
}
}
say "...total number of sexy #{k}-tuple primes = #{groups.len.commify}"
say "...where last 5 tuples are: #{groups.last(5).map{'('+.join(' ')+')'}.join(' ')}\n"
}
var unsexy_primes = primes.grep {|p| is_prime(p+6) || is_prime(p-6) -> not }
say "...total number of unsexy primes = #{unsexy_primes.len.commify}"
say "...where last 10 unsexy primes are: #{unsexy_primes.last(10)}"
```
{{out}}
```txt
Total number of primes <= 1,000,035 is 78,500.
Sexy k-tuple primes <= 1,000,035:
...total number of sexy 2-tuple primes = 16,386
...where last 5 tuples are: (999371 999377) (999431 999437) (999721 999727) (999763 999769) (999953 999959)
...total number of sexy 3-tuple primes = 2,900
...where last 5 tuples are: (997427 997433 997439) (997541 997547 997553) (998071 998077 998083) (998617 998623 998629) (998737 998743 998749)
...total number of sexy 4-tuple primes = 325
...where last 5 tuples are: (977351 977357 977363 977369) (983771 983777 983783 983789) (986131 986137 986143 986149) (990371 990377 990383 990389) (997091 997097 997103 997109)
...total number of sexy 5-tuple primes = 1
...where last 5 tuples are: (5 11 17 23 29)
...total number of unsexy primes = 48,627
...where last 10 unsexy primes are: [999853, 999863, 999883, 999907, 999917, 999931, 999961, 999979, 999983, 1000003]
```
## zkl
Using GMP (GNU Multiple Precision Arithmetic Library, probabilistic
primes), because it is easy and fast to generate primes.
[[Extensible prime generator#zkl]] could be used instead.
```zkl
var [const] BI=Import("zklBigNum"); // libGMP
const N=1_000_035, M=N+24; // M allows prime group to span N, eg N=100, (97,103)
const OVR=6; // 6 if prime group can NOT span N, else 0
ps,p := Data(M+50).fill(0), BI(1); // slop at the end (for reverse wrap around)
while(p.nextPrime()<=M){ ps[p]=1 } // bitmap of primes
ns:=(N-OVR).filter('wrap(n){ 2==(ps[n] + ps[n+6]) }); # know 2 isn't, check anyway
msg(N,"sexy prime pairs",ns,5,1);
ns:=[3..N-(6+OVR),2].filter('wrap(n){ 3==(ps[n] + ps[n+6] + ps[n+12]) }); # can't be even
msg(N,"sexy triplet primes",ns,5,2);
ns:=[3..N-(12+OVR),2].filter('wrap(n){ 4==(ps[n] + ps[n+6] + ps[n+12] + ps[n+18]) }); # no evens
msg(N,"sexy quadruplet primes",ns,5,3);
ns:=[3..N-(18+OVR),2].filter('wrap(n){ 5==(ps[n] + ps[n+6] + ps[n+12] + ps[n+18] + ps[n+24]) });
msg(N,"sexy quintuplet primes",ns,1,4);
ns:=(N-OVR).filter('wrap(n){ ps[n] and 0==(ps[n-6] + ps[n+6]) }); // include 2
msg(N,"unsexy primes",ns,10,0);
fcn msg(N,s,ps,n,g){
n=n.min(ps.len()); // if the number of primes is less than n
gs:=ps[-n,*].apply('wrap(n){ [0..g*6,6].apply('+(n)) })
.pump(String,T("concat", ","),"(%s) ".fmt);
println("Number of %s less than %,d is %,d".fmt(s,N,ps.len()));
println("The last %d %s:\n %s\n".fmt(n, (n>1 and "are" or "is"), gs));
}
```
{{out}}
Number of sexy prime pairs less than 1,000,035 is 16,386
The last 5 are:
(999371,999377) (999431,999437) (999721,999727) (999763,999769) (999953,999959)
Number of sexy triplet primes less than 1,000,035 is 2,900
The last 5 are:
(997427,997433,997439) (997541,997547,997553) (998071,998077,998083) (998617,998623,998629) (998737,998743,998749)
Number of sexy quadruplet primes less than 1,000,035 is 325
The last 5 are:
(977351,977357,977363,977369) (983771,983777,983783,983789) (986131,986137,986143,986149) (990371,990377,990383,990389) (997091,997097,997103,997109)
Number of sexy quintuplet primes less than 1,000,035 is 1
The last 1 is:
(5,11,17,23,29)
Number of unsexy primes less than 1,000,035 is 48,627
The last 10 are:
(999853) (999863) (999883) (999907) (999917) (999931) (999961) (999979) (999983) (1000003)
```