Sexy primes
In mathematics, sexy primes are prime numbers that differ from each other by six.
You are encouraged to solve this task according to the task description, using any language you may know.
| This page uses content from Wikipedia. The original article was at Sexy_prime. The list of authors can be seen in the page history. As with Rosetta Code, the text of Wikipedia is available under the GNU FDL. (See links for details on variance) |
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
<lang 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
} </lang>
- Output:
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. <lang c>#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;
}</lang>
- Output:
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]
C++
<lang cpp>#include <array>
- include <iostream>
- include <vector>
- include <boost/circular_buffer.hpp>
- include "prime_sieve.hpp"
int main() {
using std::cout; using std::vector; using boost::circular_buffer; using group_buffer = circular_buffer<vector<int>>;
const int max = 1000035; const int max_group_size = 5; const int diff = 6; const int array_size = max + diff; const int max_groups = 5; const int max_unsexy = 10;
// Use Sieve of Eratosthenes to find prime numbers up to max prime_sieve sieve(array_size);
std::array<int, max_group_size> group_count{0};
vector<group_buffer> groups(max_group_size, group_buffer(max_groups));
int unsexy_count = 0;
circular_buffer<int> unsexy_primes(max_unsexy);
vector<int> group;
for (int p = 2; p < max; ++p) {
if (!sieve.is_prime(p))
continue;
if (!sieve.is_prime(p + diff) && (p - diff < 2 || !sieve.is_prime(p - diff))) {
// if p + diff and p - diff aren't prime then p can't be sexy
++unsexy_count;
unsexy_primes.push_back(p);
} else {
// find the groups of sexy primes that begin with p
group.clear();
group.push_back(p);
for (int group_size = 1; group_size < max_group_size; group_size++) {
int next_p = p + group_size * diff;
if (next_p >= max || !sieve.is_prime(next_p))
break;
group.push_back(next_p);
++group_count[group_size];
groups[group_size].push_back(group);
}
}
}
for (int size = 1; size < max_group_size; ++size) {
cout << "number of groups of size " << size + 1 << " is " << group_count[size] << '\n';
cout << "last " << groups[size].size() << " groups of size " << size + 1 << ":";
for (const vector<int>& group : groups[size]) {
cout << " (";
for (size_t i = 0; i < group.size(); ++i) {
if (i > 0)
cout << ' ';
cout << group[i];
}
cout << ")";
}
cout << "\n\n";
}
cout << "number of unsexy primes is " << unsexy_count << '\n';
cout << "last " << unsexy_primes.size() << " unsexy primes:";
for (int prime : unsexy_primes)
cout << ' ' << prime;
cout << '\n';
return 0;
}</lang>
Contents of prime_sieve.hpp: <lang cpp>#ifndef PRIME_SIEVE_HPP
- define PRIME_SIEVE_HPP
- include <algorithm>
- include <vector>
/**
* A simple implementation of the Sieve of Eratosthenes. * See https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes. */
class prime_sieve { public:
explicit prime_sieve(size_t); bool is_prime(size_t) const;
private:
std::vector<bool> is_prime_;
};
/**
* Constructs a sieve with the given limit. * * @param limit the maximum integer that can be tested for primality */
inline prime_sieve::prime_sieve(size_t limit) {
limit = std::max(size_t(3), limit);
is_prime_.resize(limit/2, true);
for (size_t p = 3; p * p <= limit; p += 2) {
if (is_prime_[p/2 - 1]) {
size_t inc = 2 * p;
for (size_t q = p * p; q <= limit; q += inc)
is_prime_[q/2 - 1] = false;
}
}
}
/**
* Returns true if the given integer is a prime number. The integer * must be less than or equal to the limit passed to the constructor. * * @param n an integer less than or equal to the limit passed to the * constructor * @return true if the integer is prime */
inline bool prime_sieve::is_prime(size_t n) const {
if (n == 2)
return true;
if (n < 2 || n % 2 == 0)
return false;
return is_prime_.at(n/2 - 1);
}
- endif</lang>
- Output:
number of groups of size 2 is 16386 last 5 groups of size 2: (999371 999377) (999431 999437) (999721 999727) (999763 999769) (999953 999959) number of groups of size 3 is 2900 last 5 groups of size 3: (997427 997433 997439) (997541 997547 997553) (998071 998077 998083) (998617 998623 998629) (998737 998743 998749) number of groups of size 4 is 325 last 5 groups of size 4: (977351 977357 977363 977369) (983771 983777 983783 983789) (986131 986137 986143 986149) (990371 990377 990383 990389) (997091 997097 997103 997109) number of groups of size 5 is 1 last 1 groups of size 5: (5 11 17 23 29) number of unsexy primes is 48627 last 10 unsexy primes: 999853 999863 999883 999907 999917 999931 999961 999979 999983 1000003
F#
This task uses Extensible Prime Generator (F#) <lang fsharp> // 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 "" </lang>
- Output:
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
<lang 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</lang>
- Output:
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
<lang 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:])
}</lang>
- Output:
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]
Haskell
Uses Library primes. https://hackage.haskell.org/package/primes (wheel sieve). <lang haskell>import Text.Printf (printf) import Data.Numbers.Primes (isPrime, primes)
type Pair = (Int, Int) type Triplet = (Int, Int, Int) type Quad = (Int, Int, Int, Int) type Quin = (Int, Int, Int, Int, Int)
type Result = ([Pair], [Triplet], [Quad], [Quin], [Int])
groups :: Int -> Result -> Result groups n r@(p, t, q, qn, u)
| isPrime n4 && isPrime n3 && isPrime n2 && isPrime n1 = (addPair, addTriplet, addQuad, addQuin, u)
| isPrime n3 && isPrime n2 && isPrime n1 = (addPair, addTriplet, addQuad, qn, u)
| isPrime n2 && isPrime n1 = (addPair, addTriplet, q, qn, u)
| isPrime n1 = (addPair, t, q, qn, u)
| not (isPrime (n+6)) && not (isPrime n1) = (p, t, q, qn, n : u)
| otherwise = r
where addPair = (n1, n) : p
addTriplet = (n2, n1, n) : t
addQuad = (n3, n2, n1, n) : q
addQuin = (n4, n3, n2, n1, n) : qn
n1 = n - 6
n2 = n - 12
n3 = n - 18
n4 = n - 24
main :: IO () main = do
printf ("Number of sexy prime pairs: %d\n" <> lastFiveText) (length pairs) (lastFive pairs)
printf ("Number of sexy prime triplets: %d\n" <> lastFiveText) (length triplets) (lastFive triplets)
printf ("Number of sexy prime quadruplets: %d\n" <> lastFiveText) (length quads) (lastFive quads)
printf "Number of sexy prime quintuplets: %d\n Last 1 : %s\n\n" (length quins) (show $ last quins)
printf "Number of unsexy primes: %d\n Last 10: %s\n\n" (length unsexy) (show $ drop (length unsexy - 10) unsexy)
where (pairs, triplets, quads, quins, unsexy) = foldr groups ([], [], [], [], []) $ takeWhile (< 1000035) primes
lastFive xs = show $ drop (length xs - 5) xs
lastFiveText = " Last 5 : %s\n\n"</lang>
- Output:
Number of sexy prime pairs: 16386 Last 5 : [(999371,999377),(999431,999437),(999721,999727),(999763,999769),(999953,999959)] Number of sexy prime triplets: 2900 Last 5 : [(997427,997433,997439),(997541,997547,997553),(998071,998077,998083),(998617,998623,998629),(998737,998743,998749)] Number of sexy prime quadruplets: 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 Last 1 : [(5,11,17,23,29)] Number of unsexy primes: 48627 Last 10: [999853,999863,999883,999907,999917,999931,999961,999979,999983,1000003]
Slight variation which only holds on to the display results. Does not perform any better than above though. <lang haskell>import Control.Lens (_1, _2, _3, _4, _5, over) import Data.Numbers.Primes (isPrime, primes) import Text.Printf (printf)
type Pair = (Int, [(Int, Int)]) type Triplet = (Int, [(Int, Int, Int)]) type Quad = (Int, [(Int, Int, Int, Int)]) type Quin = (Int, [(Int, Int, Int, Int, Int)]) type UnSexy = (Int, [Int])
type Result = (Pair, Triplet, Quad, Quin, UnSexy) type DisplayResult = ((Int, String), (Int, String), (Int, String), (Int, String), (Int, String))
emptyGroups :: Result emptyGroups = ((0, []), (0, []), (0, []), (0, []), (0, []))
groups :: Result -> Int -> Result groups r@(pr, tt, qd, qn, un) n
| isPrime n4 && isPrime n3 && isPrime n2 && isPrime n1 = (addPair pr, addTriplet tt, addQuad qd, addQuin qn, un) | isPrime n3 && isPrime n2 && isPrime n1 = (addPair pr, addTriplet tt, addQuad qd, qn, un) | isPrime n2 && isPrime n1 = (addPair pr, addTriplet tt, qd, qn, un) | isPrime n1 = (addPair pr, tt, qd, qn, un) | not (isPrime (n+6)) && not (isPrime n1) = (pr, tt, qd, qn, addUnSexy un) | otherwise = r where n1 = n-6 n2 = n-12 n3 = n-18 n4 = n-24
addPair = over _1 succ . over _2 (take 5 . (:) (n1, n)) addTriplet = over _1 succ . over _2 (take 5 . (:) (n2, n1, n)) addQuad = over _1 succ . over _2 (take 5 . (:) (n3, n2, n1, n)) addQuin = over _1 succ . over _2 (take 1 . (:) (n4, n3, n2, n1, n)) addUnSexy = over _1 succ . over _2 (take 10 . (:) n)
display :: Result -> DisplayResult display = over _1 f . over _2 f . over _3 f . over _4 f . over _5 f
where f :: Show b => (a, [b]) -> (a, String) f = over _2 show . over _2 reverse
main :: IO () main = do
let (pairs, triplets, quads, quins, unsexy) = collectGroups primes uncurry (printf "Number of sexy prime pairs: %d\n Last 5 : %s\n\n") pairs uncurry (printf "Number of sexy prime triplets: %d\n Last 5 : %s\n\n") triplets uncurry (printf "Number of sexy prime quadruplets: %d\n Last 5 : %s\n\n") quads uncurry (printf "Number of sexy prime quintuplets: %d\n Last 1 : %s\n\n") quins uncurry (printf "Number of unsexy primes: %d\n Last 10: %s\n\n") unsexy where collectGroups = display . foldl groups emptyGroups . takeWhile (< 1000035)
</lang>
Java
<lang java> import java.util.ArrayList; import java.util.List;
public class SexyPrimes {
public static void main(String[] args) {
sieve();
int pairs = 0;
List<String> pairList = new ArrayList<>();
int triples = 0;
List<String> tripleList = new ArrayList<>();
int quadruplets = 0;
List<String> quadrupletList = new ArrayList<>();
int unsexyCount = 1; // 2 (the even prime) not found in tests below.
List<String> unsexyList = new ArrayList<>();
for ( int i = 3 ; i < MAX ; i++ ) {
if ( i-6 >= 3 && primes[i-6] && primes[i] ) {
pairs++;
pairList.add((i-6) + " " + i);
if ( pairList.size() > 5 ) {
pairList.remove(0);
}
}
else if ( i < MAX-2 && primes[i] && ! (i+6<MAX && primes[i] && primes[i+6])) {
unsexyCount++;
unsexyList.add("" + i);
if ( unsexyList.size() > 10 ) {
unsexyList.remove(0);
}
}
if ( i-12 >= 3 && primes[i-12] && primes[i-6] && primes[i] ) {
triples++;
tripleList.add((i-12) + " " + (i-6) + " " + i);
if ( tripleList.size() > 5 ) {
tripleList.remove(0);
}
}
if ( i-16 >= 3 && primes[i-18] && primes[i-12] && primes[i-6] && primes[i] ) {
quadruplets++;
quadrupletList.add((i-18) + " " + (i-12) + " " + (i-6) + " " + i);
if ( quadrupletList.size() > 5 ) {
quadrupletList.remove(0);
}
}
}
System.out.printf("Count of sexy triples less than %,d = %,d%n", MAX, pairs);
System.out.printf("The last 5 sexy pairs:%n %s%n%n", pairList.toString().replaceAll(", ", "], ["));
System.out.printf("Count of sexy triples less than %,d = %,d%n", MAX, triples);
System.out.printf("The last 5 sexy triples:%n %s%n%n", tripleList.toString().replaceAll(", ", "], ["));
System.out.printf("Count of sexy quadruplets less than %,d = %,d%n", MAX, quadruplets);
System.out.printf("The last 5 sexy quadruplets:%n %s%n%n", quadrupletList.toString().replaceAll(", ", "], ["));
System.out.printf("Count of unsexy primes less than %,d = %,d%n", MAX, unsexyCount);
System.out.printf("The last 10 unsexy primes:%n %s%n%n", unsexyList.toString().replaceAll(", ", "], ["));
}
private static int MAX = 1_000_035; private static boolean[] primes = new boolean[MAX];
private static final void sieve() {
// primes
for ( int i = 2 ; i < MAX ; i++ ) {
primes[i] = true;
}
for ( int i = 2 ; i < MAX ; i++ ) {
if ( primes[i] ) {
for ( int j = 2*i ; j < MAX ; j += i ) {
primes[j] = false;
}
}
}
}
} </lang>
- Output:
Count of sexy triples less than 1,000,035 = 16,386 The last 5 sexy pairs: [999371 999377], [999431 999437], [999721 999727], [999763 999769], [999953 999959] Count of sexy triples less than 1,000,035 = 2,900 The last 5 sexy triples: [997427 997433 997439], [997541 997547 997553], [998071 998077 998083], [998617 998623 998629], [998737 998743 998749] Count of sexy quadruplets less than 1,000,035 = 325 The last 5 sexy quadruplets: [977351 977357 977363 977369], [983771 983777 983783 983789], [986131 986137 986143 986149], [990371 990377 990383 990389], [997091 997097 997103 997109] Count of unsexy primes less than 1,000,035 = 48,627 The last 10 unsexy primes: [999853], [999863], [999883], [999907], [999917], [999931], [999961], [999979], [999983], [1000003]
Julia
<lang 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) </lang>
- 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
<lang scala>// 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))
}</lang>
- 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]
Lua
<lang lua>local N = 1000035
-- FUNCS: local function T(t) return setmetatable(t, {__index=table}) end table.filter = function(t,f) local s=T{} for _,v in ipairs(t) do if f(v) then s[#s+1]=v end end return s end table.map = function(t,f,...) local s=T{} for _,v in ipairs(t) do s[#s+1]=f(v,...) end return s end table.lastn = function(t,n) local s=T{} n=n>#t and #t or n for i = 1,n do s[i]=t[#t-n+i] end return s end table.each = function(t,f,...) for _,v in ipairs(t) do f(v,...) end end
-- PRIMES: local sieve, primes = {false}, T{} for i = 2,N+6 do sieve[i]=true end for i = 2,N+6 do if sieve[i] then for j=i*i,N+6,i do sieve[j]=nil end end end for i = 2,N+6 do if sieve[i] then primes[#primes+1]=i end end
-- TASKS: local sexy, name = { primes }, { "primes", "pairs", "triplets", "quadruplets", "quintuplets" } local function sexy2str(v,n) local s=T{} for i=1,n do s[i]=v+(i-1)*6 end return "("..s:concat(" ")..")" end for i = 2, 5 do
sexy[i] = sexy[i-1]:filter(function(v) return v+(i-1)*6<N and sieve[v+(i-1)*6] end)
print(#sexy[i] .. " " .. name[i] .. ", ending with: " .. sexy[i]:lastn(5):map(sexy2str,i):concat(" "))
end local unsexy = primes:filter(function(v) return not (v>=N or sieve[v-6] or sieve[v+6]) end) print(#unsexy .. " unsexy, ending with: " ..unsexy:lastn(10):concat(" "))</lang>
- Output:
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) 48627 unsexy, ending with: 999853 999863 999883 999907 999917 999931 999961 999979 999983 1000003
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. <lang pascal>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.</lang>
- 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
We will use the prime iterator and primality test from the ntheory module.
<lang perl>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";</lang>
- Output:
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 Prime 11-tuplets.
<lang perl>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</lang>
Phix
<lang 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)</lang>
- Output:
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}}
Prolog
<lang prolog>sexy_prime_group(1, N, _, [N]):-
is_prime(N), !.
sexy_prime_group(Size, N, Limit, [N|Group]):-
is_prime(N), N1 is N + 6, N1 =< Limit, S1 is Size - 1, sexy_prime_group(S1, N1, Limit, Group).
print_sexy_prime_groups(Size, Limit):-
findall(G, (is_prime(P), P =< Limit, sexy_prime_group(Size, P, Limit, G)), Groups),
length(Groups, Len),
writef('Number of groups of size %t is %t\n', [Size, Len]),
last_n(Groups, 5, Len, Last, Last_len),
writef('Last %t groups of size %t: %t\n\n', [Last_len, Size, Last]).
last_n([], _, L, [], L):-!. last_n([_|List], Max, Length, Last, Last_len):-
Max < Length, !, Len1 is Length - 1, last_n(List, Max, Len1, Last, Last_len).
last_n([E|List], Max, Length, [E|Last], Last_len):-
last_n(List, Max, Length, Last, Last_len).
unsexy(P):-
P1 is P + 6, \+is_prime(P1), P2 is P - 6, \+is_prime(P2).
main(Limit):-
Max is Limit + 6,
find_prime_numbers(Max),
print_sexy_prime_groups(2, Limit),
print_sexy_prime_groups(3, Limit),
print_sexy_prime_groups(4, Limit),
print_sexy_prime_groups(5, Limit),
findall(P, (is_prime(P), P =< Limit, unsexy(P)), Unsexy),
length(Unsexy, Count),
writef('Number of unsexy primes is %t\n', [Count]),
last_n(Unsexy, 10, Count, Last10, _),
writef('Last 10 unsexy primes: %t', [Last10]).
main:-
main(1000035).</lang>
Module for finding prime numbers up to some limit: <lang prolog>:- module(prime_numbers, [find_prime_numbers/1, is_prime/1]).
- - dynamic is_prime/1.
find_prime_numbers(N):-
retractall(is_prime(_)), assertz(is_prime(2)), init_sieve(N, 3), sieve(N, 3).
init_sieve(N, P):-
P > N, !.
init_sieve(N, P):-
assertz(is_prime(P)), Q is P + 2, init_sieve(N, Q).
sieve(N, P):-
P * P > N, !.
sieve(N, P):-
is_prime(P), !, S is P * P, cross_out(S, N, P), Q is P + 2, sieve(N, Q).
sieve(N, P):-
Q is P + 2, sieve(N, Q).
cross_out(S, N, _):-
S > N, !.
cross_out(S, N, P):-
retract(is_prime(S)), !, Q is S + 2 * P, cross_out(Q, N, P).
cross_out(S, N, P):-
Q is S + 2 * P, cross_out(Q, N, P).</lang>
- Output:
Number of groups of size 2 is 16386 Last 5 groups of size 2: [[999371,999377],[999431,999437],[999721,999727],[999763,999769],[999953,999959]] Number of groups of size 3 is 2900 Last 5 groups of size 3: [[997427,997433,997439],[997541,997547,997553],[998071,998077,998083],[998617,998623,998629],[998737,998743,998749]] Number of groups of size 4 is 325 Last 5 groups of size 4: [[977351,977357,977363,977369],[983771,983777,983783,983789],[986131,986137,986143,986149],[990371,990377,990383,990389],[997091,997097,997103,997109]] Number of groups of size 5 is 1 Last 1 groups of size 5: [[5,11,17,23,29]] Number of unsexy primes is 48627 Last 10 unsexy primes: [999853,999863,999883,999907,999917,999931,999961,999979,999983,1000003]
Python
Imperative Style
<lang python>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)</lang>
- Output:
"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
This task uses Extensible_prime_generator#210-wheel_postponed_incremental_sieve <lang python>
- 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) </lang>
- Output:
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
Raku
(formerly Perl 6)
<lang perl6>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 }</lang>
- Output:
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)
REXX
<lang 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</lang>
- output 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
Ring
<lang ring> load "stdlib.ring"
primes = [] for n = 1 to 100
if isprime(n)
add(primes,n)
ok
next
see "Sexy prime pairs under 100:" + nl + nl for n = 1 to len(primes)-1
for m = n + 1 to len(primes)
if primes[m] - primes[n] = 6
see "(" + primes[n] + " " + primes[m] + ")" + nl
ok
next
next see nl
see "Sexy prime triplets under 100:" + nl +nl for n = 1 to len(primes)-2
for m = n + 1 to len(primes)-1
for x = m + 1 to len(primes)
bool1 = (primes[m] - primes[n] = 6)
bool2 = (primes[x] - primes[m] = 6)
bool = bool1 and bool2
if bool
see "(" + primes[n] + " " + primes[m] + " " + primes[x] + ")" + nl
ok
next
next
next see nl
see "Sexy prime quadruplets under 100:" + nl + nl for n = 1 to len(primes)-3
for m = n + 1 to len(primes)-2
for x = m + 1 to len(primes)-1
for y = m + 1 to len(primes)
bool1 = (primes[m] - primes[n] = 6)
bool2 = (primes[x] - primes[m] = 6)
bool3 = (primes[y] - primes[x] = 6)
bool = bool1 and bool2 and bool3
if bool
see "(" + primes[n] + " " + primes[m] + " " + primes[x] + " " + primes[y] + ")" + nl
ok
next
next
next
next see nl
see "Sexy prime quintuplets under 100:" + nl + nl for n = 1 to len(primes)-4
for m = n + 1 to len(primes)-3
for x = m + 1 to len(primes)-2
for y = m + 1 to len(primes)-1
for z = y + 1 to len(primes)
bool1 = (primes[m] - primes[n] = 6)
bool2 = (primes[x] - primes[m] = 6)
bool3 = (primes[y] - primes[x] = 6)
bool4 = (primes[z] - primes[y] = 6)
bool = bool1 and bool2 and bool3 and bool4
if bool
see "(" + primes[n] + " " + primes[m] + " " + primes[x] + " " +
primes[y] + " " + primes[z] + ")" + nl
ok
next
next
next
next
next </lang> Output:
Sexy prime pairs under 100: (5 11) (7 13) (11 17) (13 19) (17 23) (23 29) (31 37) (37 43) (41 47) (47 53) (53 59) (61 67) (67 73) (73 79) (83 89) Sexy prime triplets under 100: (5 11 17) (7 13 19) (11 17 23) (17 23 29) (31 37 43) (41 47 53) (47 53 59) (61 67 73) (67 73 79) Sexy prime quadruplets under 100: (5 11 17 23) (11 17 23 29) (41 47 53 59) (61 67 73 79) Sexy prime quintuplets under 100: (5 11 17 23 29)
Ruby
<lang 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" </lang>
Output:
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
Rust
<lang rust>// [dependencies] // primal = "0.2" // circular-queue = "0.2.5"
use circular_queue::CircularQueue;
fn main() {
let max = 1000035; let max_group_size = 5; let diff = 6; let max_groups = 5; let max_unsexy = 10;
let sieve = primal::Sieve::new(max + diff); let mut group_count = vec![0; max_group_size]; let mut unsexy_count = 0; let mut groups = Vec::new(); let mut unsexy_primes = CircularQueue::with_capacity(max_unsexy);
for _ in 0..max_group_size {
groups.push(CircularQueue::with_capacity(max_groups));
}
for p in sieve.primes_from(2).take_while(|x| *x < max) {
if !sieve.is_prime(p + diff) && (p < diff + 2 || !sieve.is_prime(p - diff)) {
unsexy_count += 1;
unsexy_primes.push(p);
} else {
let mut group = Vec::new();
group.push(p);
for group_size in 1..max_group_size {
let next = p + group_size * diff;
if next >= max || !sieve.is_prime(next) {
break;
}
group.push(next);
group_count[group_size] += 1;
groups[group_size].push(group.clone());
}
}
}
for size in 1..max_group_size {
println!(
"Number of groups of size {} is {}",
size + 1,
group_count[size]
);
println!("Last {} groups of size {}:", groups[size].len(), size + 1);
println!(
"{}\n",
groups[size]
.asc_iter()
.map(|g| format!("({})", to_string(&mut g.iter())))
.collect::<Vec<String>>()
.join(", ")
);
}
println!("Number of unsexy primes is {}", unsexy_count);
println!("Last {} unsexy primes:", unsexy_primes.len());
println!("{}", to_string(&mut unsexy_primes.asc_iter()));
}
fn to_string<T: ToString>(iter: &mut dyn std::iter::Iterator<Item = T>) -> String {
iter.map(|n| n.to_string())
.collect::<Vec<String>>()
.join(", ")
}</lang>
- Output:
Number of groups of size 2 is 16386 Last 5 groups of size 2: (999371, 999377), (999431, 999437), (999721, 999727), (999763, 999769), (999953, 999959) Number of groups of size 3 is 2900 Last 5 groups of size 3: (997427, 997433, 997439), (997541, 997547, 997553), (998071, 998077, 998083), (998617, 998623, 998629), (998737, 998743, 998749) Number of groups of size 4 is 325 Last 5 groups of size 4: (977351, 977357, 977363, 977369), (983771, 983777, 983783, 983789), (986131, 986137, 986143, 986149), (990371, 990377, 990383, 990389), (997091, 997097, 997103, 997109) Number of groups of size 5 is 1 Last 1 groups of size 5: (5, 11, 17, 23, 29) Number of unsexy primes is 48627 Last 10 unsexy primes: 999853, 999863, 999883, 999907, 999917, 999931, 999961, 999979, 999983, 1000003
Scala
<lang scala>/* We could reduce the number of functions through a polymorphism since we're trying to retrieve sexy N-tuples (pairs, triplets etc...)
but one practical solution would be to use the Shapeless library for this purpose; here we only use built-in Scala packages. */
object SexyPrimes {
/** Check if an input number is prime or not*/ def isPrime(n: Int): Boolean = ! ((2 until n-1) exists (n % _ == 0)) && n > 1
/** Retrieve pairs of sexy primes given a list of Integers*/
def getSexyPrimesPairs (primes : List[Int]) = {
primes
.map(n => if(primes.contains(n+6)) (n, n+6))
.filter(p => p != ())
.map{ case (a,b) => (a.toString.toInt, b.toString.toInt)}
}
/** Retrieve triplets of sexy primes given a list of Integers*/
def getSexyPrimesTriplets (primes : List[Int]) = {
primes
.map(n => if(
primes.contains(n+6) && primes.contains(n+12))
(n, n+6, n+12)
)
.filter(p => p != ())
.map{ case (a,b,c) => (a.toString.toInt, b.toString.toInt, c.toString.toInt)}
}
/** Retrieve quadruplets of sexy primes given a list of Integers*/
def getSexyPrimesQuadruplets (primes : List[Int]) = {
primes
.map(n => if(
primes.contains(n+6) && primes.contains(n+12) && primes.contains(n+18))
(n, n+6, n+12, n+18)
)
.filter(p => p != ())
.map{ case (a,b,c,d) => (a.toString.toInt, b.toString.toInt, c.toString.toInt, d.toString.toInt)}
}
/** Retrieve quintuplets of sexy primes given a list of Integers*/
def getSexyPrimesQuintuplets (primes : List[Int]) = {
primes
.map(n => if (
primes.contains(n+6) && primes.contains(n+12) && primes.contains(n+18) && primes.contains(n + 24))
(n, n + 6, n + 12, n + 18, n + 24)
)
.filter(p => p != ())
.map { case (a, b, c, d, e) => (a.toString.toInt, b.toString.toInt, c.toString.toInt, d.toString.toInt, e.toString.toInt) }
}
/** Retrieve all unsexy primes between 1 and a given limit from an input list of Integers*/
def removeOutsideSexyPrimes( l : List[Int], limit : Int) : List[Int] = {
l.filter(n => !isPrime(n+6) && n+6 < limit)
}
def main(args: Array[String]): Unit = {
val limit = 1000035
val l = List.range(1,limit)
val primes = l.filter( n => isPrime(n))
val sexyPairs = getSexyPrimesPairs(primes)
println("Number of sexy pairs : " + sexyPairs.size)
println("5 last sexy pairs : " + sexyPairs.takeRight(5))
val primes2 = sexyPairs.flatMap(t => List(t._1, t._2)).distinct.sorted
val sexyTriplets = getSexyPrimesTriplets(primes2)
println("Number of sexy triplets : " + sexyTriplets.size)
println("5 last sexy triplets : " + sexyTriplets.takeRight(5))
val primes3 = sexyTriplets.flatMap(t => List(t._1, t._2, t._3)).distinct.sorted
val sexyQuadruplets = getSexyPrimesQuadruplets(primes3)
println("Number of sexy quadruplets : " + sexyQuadruplets.size)
println("5 last sexy quadruplets : " + sexyQuadruplets.takeRight(5))
val primes4 = sexyQuadruplets.flatMap(t => List(t._1, t._2, t._3, t._4)).distinct.sorted
val sexyQuintuplets = getSexyPrimesQuintuplets(primes4)
println("Number of sexy quintuplets : " + sexyQuintuplets.size)
println("The last sexy quintuplet : " + sexyQuintuplets.takeRight(10))
val sexyPrimes = primes2.toSet
val unsexyPrimes = removeOutsideSexyPrimes( primes.toSet.diff((sexyPrimes)).toList.sorted, limit)
println("Number of unsexy primes : " + unsexyPrimes.size)
println("10 last unsexy primes : " + unsexyPrimes.takeRight(10))
}
} </lang>
- Output:
Number of sexy pairs : 16386 5 last sexy pairs : List((999371,999377), (999431,999437), (999721,999727), (999763,999769), (999953,999959)) Number of sexy triplets : 2900 5 last sexy triplets : List((997427,997433,997439), (997541,997547,997553), (998071,998077,998083), (998617,998623,998629), (998737,998743,998749)) Number of sexy quadruplets : 325 5 last sexy quadruplets : List((977351,977357,977363,977369), (983771,983777,983783,983789), (986131,986137,986143,986149), (990371,990377,990383,990389), (997091,997097,997103,997109)) Number of sexy quintuplets : 1 The last sexy quintuplet : List((5,11,17,23,29)) Number of unsexy primes : 48627 10 last unsexy primes : List(999853, 999863, 999883, 999907, 999917, 999931, 999961, 999979, 999983, 1000003)
Sidef
<lang 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)}"</lang>
- Output:
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]
Wren
<lang ecmascript>import "/fmt" for Fmt import "/math" for Int
var printHelper = Fn.new { |cat, le, lim, max|
var cle = Fmt.commatize(le)
var clim = Fmt.commatize(lim)
if (cat != "unsexy primes") cat = "sexy prime " + cat
System.print("Number of %(cat) less than %(clim) = %(cle)")
var last = (le < max) ? le : max
var verb = (last == 1) ? "is" : "are"
return [le, last, verb]
}
var lim = 1000035 var sv = Int.primeSieve(lim-1, false) var pairs = [] var trips = [] var quads = [] var quins = [] var unsexy = [2, 3] var i = 3 while (i < lim) {
if (i > 5 && i < lim-6 && !sv[i] && sv[i-6] && sv[i+6]) {
unsexy.add(i)
} else {
if (i < lim-6 && !sv[i] && !sv[i+6]) {
pairs.add([i, i+6])
if (i < lim-12 && !sv[i+12]) {
trips.add([i, i+6, i+12])
if (i < lim-18 && !sv[i+18]) {
quads.add([i, i+6, i+12, i+18])
if (i < lim-24 && !sv[i+24]) {
quins.add([i, i+6, i+12, i+18, i+24])
}
}
}
}
}
i = i + 2
} var le var n var verb var unwrap = Fn.new { |t|
le = t[0] n = t[1] verb = t[2]
}
unwrap.call(printHelper.call("pairs", pairs.count, lim, 5)) System.print("The last %(n) %(verb):\n %(pairs[le-n..-1])\n")
unwrap.call(printHelper.call("triplets", trips.count, lim, 5)) System.print("The last %(n) %(verb):\n %(trips[le-n..-1])\n")
unwrap.call(printHelper.call("quadruplets", quads.count, lim, 5)) System.print("The last %(n) %(verb):\n %(quads[le-n..-1])\n")
unwrap.call(printHelper.call("quintuplets", quins.count, lim, 5)) System.print("The last %(n) %(verb):\n %(quins[le-n..-1])\n")
unwrap.call(printHelper.call("unsexy primes", unsexy.count, lim, 10)) System.print("The last %(n) %(verb):\n %(unsexy[le-n..-1])\n")</lang>
- Output:
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]
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. <lang 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));
}</lang>
- Output:
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)