Ascending primes

Generate and show all primes with strictly ascending decimal digits.

Task
Ascending primes
You are encouraged to solve this task according to the task description, using any language you may know.

Aside: Try solving without peeking at existing solutions. I had a weird idea for generating a prime sieve faster, which needless to say didn't pan out. The solution may be p(r)etty trivial but generating them quickly is at least mildly interesting. Tip: filtering all 7,027,260 primes below 123,456,789 probably won't kill you, but there is at least one significantly better and much faster way, needing a mere 511 odd/prime tests.


See also


Related


ALGOL 68Edit

Uses Pete's hint to enumerate the 512 possible numbers.
The numbers are generated in order of the first digit, so we have to sort them. As there are only 512 possible numbers to consider, it doesn't attempt the optimisation that the final digit can't be 4, 6 or 8 and can only be 2 or 5 if it is the only digit (also, I always forget that can't be even thing...).

Works with: ALGOL 68G version Any - tested with release 2.8.3.win32
Library: ALGOL 68-rows
BEGIN # find all primes with strictly increasing digits                      #
    PR read "primes.incl.a68" PR                   # include prime utilities #
    PR read "rows.incl.a68"   PR                   # include array utilities #
    [ 1 : 512 ]INT primes;         # there will be at most 512 (2^9) primes  #
    INT p count := 0;                        # number of primes found so far #
    FOR d1 FROM 0 TO 1 DO
        INT n1 = d1;
        FOR d2 FROM 0 TO 1 DO
            INT n2 = IF d2 = 1 THEN ( n1 * 10 ) + 2 ELSE n1 FI;
            FOR d3 FROM 0 TO 1 DO
                INT n3 = IF d3 = 1 THEN ( n2 * 10 ) + 3 ELSE n2 FI;
                FOR d4 FROM 0 TO 1 DO
                    INT n4 = IF d4 = 1 THEN ( n3 * 10 ) + 4 ELSE n3 FI;
                    FOR d5 FROM 0 TO 1 DO
                        INT n5 = IF d5 = 1 THEN ( n4 * 10 ) + 5 ELSE n4 FI;
                        FOR d6 FROM 0 TO 1 DO
                            INT n6 = IF d6 = 1 THEN ( n5 * 10 ) + 6 ELSE n5 FI;
                            FOR d7 FROM 0 TO 1 DO
                                INT n7 = IF d7 = 1 THEN ( n6 * 10 ) + 7 ELSE n6 FI;
                                FOR d8 FROM 0 TO 1 DO
                                    INT n8 = IF d8 = 1 THEN ( n7 * 10 ) + 8 ELSE n7 FI;
                                    FOR d9 FROM 0 TO 1 DO
                                        INT n9 = IF d9 = 1 THEN ( n8 * 10 ) + 9 ELSE n8 FI;
                                        IF n9 > 0 THEN
                                            IF is probably prime( n9 ) THEN
                                                # have a prime with strictly ascending digits #
                                                primes[ p count +:= 1 ] := n9
                                            FI
                                        FI
                                    OD
                                OD
                            OD
                        OD
                    OD
                OD
            OD
        OD
    OD;
    QUICKSORT primes FROMELEMENT 1 TOELEMENT p count;     # sort the primes #
    FOR i TO p count DO                                # display the primes #
        print( ( "  ", whole( primes[ i ], -8 ) ) );
        IF i MOD 10 = 0 THEN print( ( newline ) ) FI
    OD
END
Output:
         2         3         5         7        13        17        19        23        29        37
        47        59        67        79        89       127       137       139       149       157
       167       179       239       257       269       347       349       359       367       379
       389       457       467       479       569      1237      1249      1259      1279      1289
      1367      1459      1489      1567      1579      1789      2347      2357      2389      2459
      2467      2579      2689      2789      3457      3467      3469      4567      4679      4789
      5689     12347     12379     12457     12479     12569     12589     12689     13457     13469
     13567     13679     13789     15679     23459     23567     23689     23789     25679     34589
     34679    123457    123479    124567    124679    125789    134789    145679    234589    235679
    235789    245789    345679    345689   1234789   1235789   1245689   1456789  12356789  23456789

ArturoEdit

ascending?: function [x][
    initial: digits x
    and? [equal? sort initial initial][equal? size initial size unique initial]
]

candidates: select (1..1456789) ++ [
    12345678, 12345679, 12345689, 12345789, 12346789,
    12356789, 12456789, 13456789, 23456789, 123456789 
] => prime?

ascendingNums: select candidates => ascending?

loop split.every:10 ascendingNums 'nums [
    print map nums 'num -> pad to :string num 10
]
Output:
         2          3          5          7         13         17         19         23         29         37 
        47         59         67         79         89        127        137        139        149        157 
       167        179        239        257        269        347        349        359        367        379 
       389        457        467        479        569       1237       1249       1259       1279       1289 
      1367       1459       1489       1567       1579       1789       2347       2357       2389       2459 
      2467       2579       2689       2789       3457       3467       3469       4567       4679       4789 
      5689      12347      12379      12457      12479      12569      12589      12689      13457      13469 
     13567      13679      13789      15679      23459      23567      23689      23789      25679      34589 
     34679     123457     123479     124567     124679     125789     134789     145679     234589     235679 
    235789     245789     345679     345689    1234789    1235789    1245689    1456789   12356789   23456789

AWKEdit

# syntax: GAWK -f ASCENDING_PRIMES.AWK
BEGIN {
    start = 1
    stop = 23456789
    for (i=start; i<=stop; i++) {
      if (is_prime(i)) {
        primes++
        leng = length(i)
        flag = 1
        for (j=1; j<leng; j++) {
          if (substr(i,j,1) >= substr(i,j+1,1)) {
            flag = 0
            break
          }
        }
        if (flag) {
          printf("%9d%1s",i,++count%10?"":"\n")
        }
      }
    }
    printf("\n%d-%d: %d primes, %d ascending primes\n",start,stop,primes,count)
    exit(0)
}
function is_prime(n,  d) {
    d = 5
    if (n < 2) { return(0) }
    if (n % 2 == 0) { return(n == 2) }
    if (n % 3 == 0) { return(n == 3) }
    while (d*d <= n) {
      if (n % d == 0) { return(0) }
      d += 2
      if (n % d == 0) { return(0) }
      d += 4
    }
    return(1)
}
Output:
        2         3         5         7        13        17        19        23        29        37
       47        59        67        79        89       127       137       139       149       157
      167       179       239       257       269       347       349       359       367       379
      389       457       467       479       569      1237      1249      1259      1279      1289
     1367      1459      1489      1567      1579      1789      2347      2357      2389      2459
     2467      2579      2689      2789      3457      3467      3469      4567      4679      4789
     5689     12347     12379     12457     12479     12569     12589     12689     13457     13469
    13567     13679     13789     15679     23459     23567     23689     23789     25679     34589
    34679    123457    123479    124567    124679    125789    134789    145679    234589    235679
   235789    245789    345679    345689   1234789   1235789   1245689   1456789  12356789  23456789

1-23456789: 1475171 primes, 100 ascending primes

CEdit

Translation of: Fortran
/*
 *  Ascending primes
 *
 *  Generate and show all primes with strictly ascending decimal digits.
 *
 *
 *  Solution
 *
 *  We only consider positive numbers in the range 1 to 123456789. We would
 *  get 7027260 primes, because there are so many primes smaller than 123456789
 *  (see also Wolfram Alpha).On the other hand, there are only 511 distinct
 *  nonzero positive integers having their digits arranged in ascending order.
 *  Therefore, it is better to start with numbers that have properly arranged
 *  digitsand then check if they are prime numbers.The method of generating
 *  a sequence of such numbers is not indifferent.We want this sequence to be
 *  monotonically increasing, because then additional sorting of results will
 *  be unnecessary. It turns out that by using a queue we can easily get the
 *  desired effect. Additionally, the algorithm then does not use recursion
 *  (although the program probably does not have to comply with the MISRA
 *  standard). The problem to be solved is the queue size, the a priori
 *  assumption that 1000 is good enough, but a bit magical.
 */

#include <stdio.h>
#include <stdlib.h>
#include <stdbool.h>
#include <math.h>

#if UINT_MAX < 123456789
#error "we need at least 9 decimal digits (32-bit integers)"
#endif


#define MAXSIZE 1000

unsigned queue[MAXSIZE];
unsigned primes[MAXSIZE];

unsigned begin = 0;
unsigned end = 0;
unsigned n = 0;


bool isPrime(unsigned n)
{
    if (n == 2)
    {
        return true;
    }
    if (n == 1 || n % 2 == 0)
    {
        return false;
    }
    unsigned root = sqrt(n);
    for (unsigned k = 3; k <= root; k += 2)
    {
        if (n % k == 0)
        {
            return false;
        }
    }
    return true;
}


int main(int argc, char argv[])
{
    for (int k = 1; k <= 9; k++)
    {
        queue[end++] = k;
    }

    while (begin < end)
    {
        int value = queue[begin++];
        if (isPrime(value))
        {
            primes[n++] = value;
        }
        for (int k = value % 10 + 1; k <= 9; k++)
        {
            queue[end++] = value * 10 + k;
        }
    }

    for (int k = 0; k < n; k++)
    {
        printf("%u ", primes[k]);
    }
 
    return EXIT_SUCCESS;
}
Output:
2 3 5 7 13 17 19 23 29 37 47 59 67 79 89 127 137 139 149 157 167 179 239 257 269 347 349 359 367 379 389 457 467 479 569 1237 1249 1259 1279 1289 1367 1459 1489 1567 1579 1789 2347 2357 2389 2459 2467 2579 2689 2789 3457 3467 3469 4567 4679 4789 5689 12347 12379 12457 12479 12569 12589 12689 13457 13469 13567 13679 13789 15679 23459 23567 23689 23789 25679 34589 34679 123457 123479 124567 124679 125789 134789 145679 234589 235679 235789 245789 345679 345689 1234789 1235789 1245689 1456789 12356789 23456789 

C++Edit

Translation of: C
/*
 *  Ascending primes
 *
 *  Generate and show all primes with strictly ascending decimal digits.
 *
 *
 *  Solution
 *
 *  We only consider positive numbers in the range 1 to 123456789. We would
 *  get 7027260 primes, because there are so many primes smaller than 123456789
 *  (see also Wolfram Alpha).On the other hand, there are only 511 distinct
 *  nonzero positive integers having their digits arranged in ascending order.
 *  Therefore, it is better to start with numbers that have properly arranged
 *  digitsand then check if they are prime numbers.The method of generating
 *  a sequence of such numbers is not indifferent.We want this sequence to be
 *  monotonically increasing, because then additional sorting of results will
 *  be unnecessary. It turns out that by using a queue we can easily get the
 *  desired effect. Additionally, the algorithm then does not use recursion
 *  (although the program probably does not have to comply with the MISRA
 *  standard). The problem to be solved is the queue size, the a priori
 *  assumption that 1000 is good enough, but a bit magical.
 */

#include <cmath>
#include <iostream>
#include <queue>
#include <vector>

using namespace std;


queue<unsigned> suspected;
vector<unsigned> primes;


bool isPrime(unsigned n)
{
    if (n == 2)
    {
        return true;
    }
    if (n == 1 || n % 2 == 0)
    {
        return false;
    }
    unsigned root = sqrt(n);
    for (unsigned k = 3; k <= root; k += 2)
    {
        if (n % k == 0)
        {
            return false;
        }
    }
    return true;
}


int main(int argc, char argv[])
{
    for (unsigned k = 1; k <= 9; k++)
    {
        suspected.push(k);
    }

    while (!suspected.empty())
    {
        int n = suspected.front();
        suspected.pop();

        if (isPrime(n))
        {
            primes.push_back(n);
        }

        //  The value of n % 10 gives the least significient digit of n
        //
        for (unsigned k = n % 10 + 1; k <= 9; k++)
        {
            suspected.push(n * 10 + k);
        }
    }

    copy(primes.begin(), primes.end(), ostream_iterator<unsigned>(cout, " "));

    return EXIT_SUCCESS;
}
Output:
2 3 5 7 13 17 19 23 29 37 47 59 67 79 89 127 137 139 149 157 167 179 239 257 269 347 349 359 367 379 389 457 467 479 569 1237 1249 1259 1279 1289 1367 1459 1489 1567 1579 1789 2347 2357 2389 2459 2467 2579 2689 2789 3457 3467 3469 4567 4679 4789 5689 12347 12379 12457 12479 12569 12589 12689 13457 13469 13567 13679 13789 15679 23459 23567 23689 23789 25679 34589 34679 123457 123479 124567 124679 125789 134789 145679 234589 235679 235789 245789 345679 345689 1234789 1235789 1245689 1456789 12356789 23456789 

C#Edit

Translation of: PHP
using System;
using System.Collections.Generic;

namespace ascendingprimes
{
    class Program
    {
        static bool isPrime(uint n)
        {
            if (n == 2)
                return true;
            if (n == 1 || n % 2 = 0)
                return false;
            uint root = (uint)Math.Sqrt(n);
            for (uint k = 3; k <= root; k += 2)
                if (n % k == 0)
                    return false;
            return true;
        }
        static void Main(string[] args)
        {
            var queue = new Queue<uint>();
            var primes = new List<uint>();

            for (uint k = 1; k <= 9; k++)
                queue.Enqueue(k);
            while(queue.Count > 0)
            {
                uint n = queue.Dequeue();
                if (isPrime(n))
                    primes.Add(n);
                for (uint k = n % 10 + 1; k <= 9; k++)
                    queue.Enqueue(n * 10 + k);
            }

            foreach (uint p in primes)
            {
                Console.Write(p);
                Console.Write(" ");
            }
            Console.WriteLine();
        }
    }
}
Output:
2 3 5 7 13 17 19 23 29 37 47 59 67 79 89 127 137 139 149 157 167 179 239 257 269 347 349 359 367 379 389 457 467 479 569 1237 1249 1259 1279 1289 1367 1459 1489 1567 1579 1789 2347 2357 2389 2459 2467 2579 2689 2789 3457 3467 3469 4567 4679 4789 5689 12347 12379 12457 12479 12569 12589 12689 13457 13469 13567 13679 13789 15679 23459 23567 23689 23789 25679 34589 34679 123457 123479 124567 124679 125789 134789 145679 234589 235679 235789 245789 345679 345689 1234789 1235789 1245689 1456789 12356789 23456789

F#Edit

This task uses Extensible Prime Generator (F#)

// Ascending primes. Nigel Galloway: April 19th., 2022
[2;3;5;7]::List.unfold(fun(n,i)->match n with []->None |_->let n=n|>List.map(fun(n,g)->[for n in n.. -1..1->(n-1,i*n+g)])|>List.concat in Some(n|>List.choose(fun(_,n)->if isPrime n then Some n else None),(n|>List.filter(fst>>(<)0),i*10)))([(2,3);(6,7);(8,9)],10)
  |>List.concat|>List.sort|>List.iter(printf "%d "); printfn ""
Output:
2 3 5 7 13 17 19 23 29 37 47 59 67 79 89 127 137 139 149 157 167 179 239 257 269 347 349 359 367 379 389 457 467 479 569 1237 1249 1259 1279 1289 1367 1459 1489 1567 1579 1789 2347 2357 2389 2459 2467 2579 2689 2789 3457 3467 3469 4567 4679 4789 5689 12347 12379 12457 12479 12569 12589 12689 13457 13469 13567 13679 13789 15679 23459 23567 23689 23789 25679 34589 34679 123457 123479 124567 124679 125789 134789 145679 234589 235679 235789 245789 345679 345689 1234789 1235789 1245689 1456789 12356789 23456789

FactorEdit

The approach taken is to check the members of the powerset of [1..9] (of which there are only 512 if you include the empty set) for primality.

Works with: Factor version 0.99 2021-06-02
USING: grouping math math.combinatorics math.functions
math.primes math.ranges prettyprint sequences sequences.extras ;

9 [1,b] all-subsets [ reverse 0 [ 10^ * + ] reduce-index ]
[ prime? ] map-filter 10 group simple-table.
Output:
2      3      5      7      13      17      19      23      29       37
47     59     67     79     89      127     137     139     149      157
167    179    239    257    269     347     349     359     367      379
389    457    467    479    569     1237    1249    1259    1279     1289
1367   1459   1489   1567   1579    1789    2347    2357    2389     2459
2467   2579   2689   2789   3457    3467    3469    4567    4679     4789
5689   12347  12379  12457  12479   12569   12589   12689   13457    13469
13567  13679  13789  15679  23459   23567   23689   23789   25679    34589
34679  123457 123479 124567 124679  125789  134789  145679  234589   235679
235789 245789 345679 345689 1234789 1235789 1245689 1456789 12356789 23456789


ForthEdit

Works with: gforth version 0.7.3


#! /usr/bin/gforth

\ Ascending primes

\ checks (by simple trial-division) whether the TOS is prime
: prime? ( n -- f )
    dup 1 <= IF
        drop false
    ELSE
        dup 2 = IF
            drop true
        ELSE
            2
            BEGIN
                2dup dup * > >r
                2dup mod 0> r>
                over and
            WHILE
                drop 1+
            REPEAT
            nip nip
        THEN
    THEN
;

: ascending-primes-aux ( n i -- )
    dup 10 = IF
        drop
        dup prime? IF
            .
        ELSE
            drop
        THEN
    ELSE
        2dup 1+ recurse                  \ do not include digit i
        swap 10 * over + swap 1+ recurse \ do include digit i
    THEN
;

\ prints all primes with strictly ascending digits
: ascending-primes ( -- )
    0 1 ascending-primes-aux cr
;

ascending-primes bye
Output:
./ascending-primes.fs 
89 7 79 67 5 59 569 5689 47 479 4789 467 4679 457 4567 3 389 37 379 367 359 349 347 3469 3467 34679 34589 3457 345689 345679 2 29 2789 269 2689 257 2579 25679 2467 2459 245789 23 239 2389 23789 23689 2357 235789 23567 235679 2347 23459 234589 23456789 19 17 179 1789 167 157 1579 1567 15679 149 1489 1459 145679 1456789 13 139 137 13789 1367 13679 13567 134789 13469 13457 1289 127 1279 12689 1259 12589 125789 12569 1249 12479 124679 12457 1245689 124567 1237 12379 1235789 12356789 12347 123479 1234789 123457

FortranEdit

! Ascending primes
!
! Generate and show all primes with strictly ascending decimal digits.
!
!
! Solution
!
! We only consider positive numbers in the range 1 to 123456789. We would get
! 7027260 primes, because there are so many primes smaller than 123456789 (see
! also Wolfram Alpha). On the other hand, there are only 511 distinct positive
! integers having their digits arranged in ascending order. Therefore, it is
! better to start with numbers that have properly arranged digits and then check
! if they are prime numbers. The method of generating a sequence of such numbers
! is not indifferent. We want this sequence to be monotonically increasing,
! because then additional sorting of results will be unnecessary. It turns out
! that by using a queue we can easily get the desired effect. Additionally, the
! algorithm then does not use recursion (although the program probably does not
! have to comply with the MISRA standard). The problem to be solved is the queue
! size, the a priori assumption that 1000 is good enough, but a bit magical.

    
program prog

    parameter (MAXSIZE = 1000)
    
    logical isprime
    dimension iqueue(MAXSIZE)
    dimension iprimes(MAXSIZE)
    
    ibegin = 1
    iend = 1
    n = 0

    do k = 1, 9
        iqueue(iend) = k
        iend = iend + 1
    end do
    
    do while (ibegin .lt. iend)
        iv = iqueue(ibegin)
        ibegin = ibegin + 1
        if (isprime(iv)) then
            n = n + 1
            iprimes(n) = iv
        end if
        lsd1 = mod(iv, 10) + 1
        if (lsd1 .le. 9) then
            do k = lsd1, 9
                iqueue(iend) = iv * 10 + k
                iend = iend + 1
            end do
        end if
    end do

    print *, (iprimes(i), i = 1, n)

end program


logical function isprime(n)
    
! Slightly improved algorithm for checking if a number is prime.
! First, we check the special cases: 0, 1, 2. Then we check whether
! the number is divisible by 2. If it is not divisible by two,
! we check whether it is divisible by odd numbers not greater than
! the square root of that number.
!
! Positive numbers only. BTW, negative numbers are prime numbers
! if their absolute values are prime numbers.

    isprime = .FALSE.
    if (n .eq. 0 .or. n .eq. 1) then
        return
    end if
    if (n .ne. 2) then
        if (mod(n, 2) .eq. 0) then
            return
        end if
        m = n**0.5
        do k = 3, m, 2
            if (mod(n, k) .eq. 0) then
                return
            end if
        end do
    end if
    isprime = .TRUE.
end function
Output:

The estimated execution time is 1.5 milliseconds on the same hardware on which the Java program was run. It should be remembered that modern CPUs do not have a constant clock speed and additionally the measured times depend on the system load with other tasks. Nevertheless, the Fortran program seems to be 4 times faster than the Java program.

           2           3           5           7          13          17
          19          23          29          37          47          59
          67          79          89         127         137         139
         149         157         167         179         239         257
         269         347         349         359         367         379
         389         457         467         479         569        1237
        1249        1259        1279        1289        1367        1459
        1489        1567        1579        1789        2347        2357
        2389        2459        2467        2579        2689        2789
        3457        3467        3469        4567        4679        4789
        5689       12347       12379       12457       12479       12569
       12589       12689       13457       13469       13567       13679
       13789       15679       23459       23567       23689       23789
       25679       34589       34679      123457      123479      124567
      124679      125789      134789      145679      234589      235679
      235789      245789      345679      345689     1234789     1235789
     1245689     1456789    12356789    23456789

GoEdit

Translation of: Wren
Library: Go-rcu

Using a generator.

package main

import (
    "fmt"
    "rcu"
    "sort"
)

var ascPrimesSet = make(map[int]bool) // avoids duplicates

func generate(first, cand, digits int) {
    if digits == 0 {
        if rcu.IsPrime(cand) {
            ascPrimesSet[cand] = true
        }
        return
    }
    for i := first; i < 10; i++ {
        next := cand*10 + i
        generate(i+1, next, digits-1)
    }
}

func main() {
    for digits := 1; digits < 10; digits++ {
        generate(1, 0, digits)
    }
    le := len(ascPrimesSet)
    ascPrimes := make([]int, le)
    i := 0
    for k := range ascPrimesSet {
        ascPrimes[i] = k
        i++
    }
    sort.Ints(ascPrimes)
    fmt.Println("There are", le, "ascending primes, namely:")
    for i := 0; i < le; i++ {
        fmt.Printf("%8d ", ascPrimes[i])
        if (i+1)%10 == 0 {
            fmt.Println()
        }
    }
}
Output:
There are 100 ascending primes, namely:
       2        3        5        7       13       17       19       23       29       37 
      47       59       67       79       89      127      137      139      149      157 
     167      179      239      257      269      347      349      359      367      379 
     389      457      467      479      569     1237     1249     1259     1279     1289 
    1367     1459     1489     1567     1579     1789     2347     2357     2389     2459 
    2467     2579     2689     2789     3457     3467     3469     4567     4679     4789 
    5689    12347    12379    12457    12479    12569    12589    12689    13457    13469 
   13567    13679    13789    15679    23459    23567    23689    23789    25679    34589 
   34679   123457   123479   124567   124679   125789   134789   145679   234589   235679 
  235789   245789   345679   345689  1234789  1235789  1245689  1456789 12356789 23456789 

JEdit

Compare with Descending primes.

Live link.

   extend=: {{ y;(1+each i._1+{.y),L:0 y }}
   $(#~ 1 p: ])10#.&>([:~.@;extend each)^:# >:i.9
100
   10 10$(#~ 1 p: ])10#.&>([:~.@;extend each)^:# >:i.9
     2      3     13     23       5       7      17      37       47       67
   127    137    347    157     257     457     167     367      467     1237
  2347   2357   3457   1367    2467    3467    1567    4567    12347    12457
 13457  13567  23567 123457  124567      19      29      59       79       89
   139    239    149    349     359     269     569     179      379      479
   389   1249   1259   1459    2459    3469    1279    1579     2579     4679
  1289   2389   1489   2689    5689    1789    2789    4789    23459    13469
 12569  12379  12479  13679   34679   15679   25679   12589    34589    12689
 23689  13789  23789 123479  124679  235679  145679  345679   234589   345689
134789 125789 235789 245789 1245689 1234789 1235789 1456789 12356789 23456789
   timex'(#~ 1 p: ])10#.&>([:~.@;extend each)^:# >:i.9' NB. seconds (take with grain of salt)
0.003818

cpu here was a 1.2ghz i3-1005g1

JavaEdit

Translation of: C++
/*
 *  Ascending primes
 *
 *  Generate and show all primes with strictly ascending decimal digits.
 *
 *
 *  Solution
 *
 *  We only consider positive numbers in the range 1 to 123456789. We would
 *  get 7027260 primes, because there are so many primes smaller than 123456789
 *  (see also Wolfram Alpha).On the other hand, there are only 511 distinct
 *  positive integers having their digits arranged in ascending order.
 *  Therefore, it is better to start with numbers that have properly arranged
 *  digits and then check if they are prime numbers.The method of generating
 *  a sequence of such numbers is not indifferent.We want this sequence to be
 *  monotonically increasing, because then additional sorting of results will
 *  be unnecessary. It turns out that by using a queue we can easily get the
 *  desired effect. Additionally, the algorithm then does not use recursion
 *  (although the program probably does not have to comply with the MISRA
 *  standard). The problem to be solved is the queue size, the a priori
 *  assumption that 1000 is good enough, but a bit magical.
 */

package example.rossetacode.ascendingprimes;

import java.util.Arrays;

public class Program implements Runnable {

    public static void main(String[] args) {
        long t1 = System.nanoTime();
        new Program().run();
        long t2 = System.nanoTime();
        System.out.println(
                "total time consumed = " + (t2 - t1) * 1E-6 + " milliseconds");
    }

    public void run() {

        final int MAX_SIZE = 1000;
        final int[] queue = new int[MAX_SIZE];
        int begin = 0;
        int end = 0;

        for (int k = 1; k <= 9; k++) {
            queue[end++] = k;
        }

        while (begin < end) {
            int n = queue[begin++];
            for (int k = n % 10 + 1; k <= 9; k++) {
                queue[end++] = n * 10 + k;
            }
        }

        // We can use a parallel stream (and then sort the results)
        // to use multiple cores.
        //
        System.out.println(Arrays.stream(queue).filter(this::isPrime).boxed().toList());
    }

    private boolean isPrime(int n) {
        if (n == 2) {
            return true;
        }
        if (n == 1 || n % 2 == 0) {
            return false;
        }
        int root = (int) Math.sqrt(n);
        for (int k = 3; k <= root; k += 2) {
            if (n % k == 0) {
                return false;
            }
        }
        return true;
    }
}
Output:
[2, 3, 5, 7, 13, 17, 19, 23, 29, 37, 47, 59, 67, 79, 89, 127, 137, 139, 149, 157, 167, 179, 239, 257, 269, 347, 349, 359, 367, 379, 389, 457, 467, 479, 569, 1237, 1249, 1259, 1279, 1289, 1367, 1459, 1489, 1567, 1579, 1789, 2347, 2357, 2389, 2459, 2467, 2579, 2689, 2789, 3457, 3467, 3469, 4567, 4679, 4789, 5689, 12347, 12379, 12457, 12479, 12569, 12589, 12689, 13457, 13469, 13567, 13679, 13789, 15679, 23459, 23567, 23689, 23789, 25679, 34589, 34679, 123457, 123479, 124567, 124679, 125789, 134789, 145679, 234589, 235679, 235789, 245789, 345679, 345689, 1234789, 1235789, 1245689, 1456789, 12356789, 23456789]
total time consumed = 4.964799999999999 milliseconds

JavaScriptEdit

Translation of: Java
<!DOCTYPE html>
<html>
<body>
  <noscript>
    No script, no fun. Turn on Javascript on.
  </noscript>

  <script>
  (()=>{

    function isPrime(n) {
      if (n == 2)
          return true;
      if (n == 1 || n % 2 == 0)
          return false;
      root = Math.sqrt(n)
      for (let k = 3; k <= root; k += 2)
          if (n % k == 0)
              return false;
      return true;
    }

    let queue = [];
    let primes = [];

    for (let k = 1; k <= 9; k++)
      queue.push(k);

    while (queue.length != 0)
    {
        let n = queue.shift();
        if (isPrime(n))
          primes.push(n);
        for (let k = n % 10 + 1; k <= 9; k++)
          queue.push(n * 10 + k);
    }

    document.writeln(primes);

  })();
  </script>

</body>
</html>
Output:
2,3,5,7,13,17,19,23,29,37,47,59,67,79,89,127,137,139,149,157,167,179,239,257,269,347,349,359,367,379,389,457,467,479,569,1237,1249,1259,1279,1289,1367,1459,1489,1567,1579,1789,2347,2357,2389,2459,2467,2579,2689,2789,3457,3467,3469,4567,4679,4789,5689,12347,12379,12457,12479,12569,12589,12689,13457,13469,13567,13679,13789,15679,23459,23567,23689,23789,25679,34589,34679,123457,123479,124567,124679,125789,134789,145679,234589,235679,235789,245789,345679,345689,1234789,1235789,1245689,1456789,12356789,23456789

jqEdit

Works with: jq

Works with gojq, the Go implementation of jq

See Erdős-primes#jq for a suitable definition of `is_prime` as used here.

# Output: the stream of ascending primes, in order
def ascendingPrimes:
  # Generate the stream of primes beginning with the digit .
  # and with strictly ascending digits, without regard to order
  def generate:
    # strings
    def g:
      . as $first
      | tonumber as $n
      | select($n <= 9)
      | $first,
        ((range($n + 1;10) | tostring | g) as $x
         | $first + $x );
    tostring | g | tonumber | select(is_prime);

  [range(1;10) | generate] | sort[];

def task:
  def lpad($len): tostring | ($len - length) as $l | (" " * $l)[:$l] + .;
  [ascendingPrimes]
  | "There are \(length) ascending primes, namely:",
    ( _nwise(10) | map(lpad(10)) | join(" ") );

task
Output:
There are 100 ascending primes, namely:
         2          3          5          7         13         17         19         23         29         37
        47         59         67         79         89        127        137        139        149        157
       167        179        239        257        269        347        349        359        367        379
       389        457        467        479        569       1237       1249       1259       1279       1289
      1367       1459       1489       1567       1579       1789       2347       2357       2389       2459
      2467       2579       2689       2789       3457       3467       3469       4567       4679       4789
      5689      12347      12379      12457      12479      12569      12589      12689      13457      13469
     13567      13679      13789      15679      23459      23567      23689      23789      25679      34589
     34679     123457     123479     124567     124679     125789     134789     145679     234589     235679
    235789     245789     345679     345689    1234789    1235789    1245689    1456789   12356789   23456789


JuliaEdit

using Combinatorics
using Primes

function ascendingprimes()
    return filter(isprime, [evalpoly(10, reverse(x))
       for x in powerset([1, 2, 3, 4, 5, 6, 7, 8, 9]) if !isempty(x)])
end

foreach(p -> print(rpad(p[2], 10), p[1] % 10 == 0 ? "\n" : ""), enumerate(ascendingprimes()))

@time ascendingprimes()
Output:
2         3         5         7         13        17        19        23        29        37        
47        59        67        79        89        127       137       139       149       157       
167       179       239       257       269       347       349       359       367       379       
389       457       467       479       569       1237      1249      1259      1279      1289      
1367      1459      1489      1567      1579      1789      2347      2357      2389      2459      
2467      2579      2689      2789      3457      3467      3469      4567      4679      4789      
5689      12347     12379     12457     12479     12569     12589     12689     13457     13469     
13567     13679     13789     15679     23459     23567     23689     23789     25679     34589
34679     123457    123479    124567    124679    125789    134789    145679    234589    235679
235789    245789    345679    345689    1234789   1235789   1245689   1456789   12356789  23456789

0.000150 seconds (2.19 k allocations: 159.078 KiB

LuaEdit

Exactly 511 calls to is_prime required.

local function is_prime(n)
  if n < 2 then return false end
  if n % 2 == 0 then return n==2 end
  if n % 3 == 0 then return n==3 end
  for f = 5, n^0.5, 6 do
    if n%f==0 or n%(f+2)==0 then return false end
  end
  return true
end

local function ascending_primes()
  local digits, candidates, primes = {1,2,3,4,5,6,7,8,9}, {0}, {}
  for i = 1, #digits do
    for j = 1, #candidates do
      local value = candidates[j] * 10 + digits[i]
      if is_prime(value) then primes[#primes+1] = value end
      candidates[#candidates+1] = value
    end
  end
  table.sort(primes)
  return primes
end

print(table.concat(ascending_primes(), ", "))
Output:
2, 3, 5, 7, 13, 17, 19, 23, 29, 37, 47, 59, 67, 79, 89, 127, 137, 139, 149, 157, 167, 179, 239, 257, 269, 347, 349, 359, 367, 379, 389, 457, 467, 479, 569, 1237, 1249, 1259, 1279, 1289, 1367, 1459, 1489, 1567, 1579, 1789, 2347, 2357, 2389, 2459, 2467, 2579, 2689, 2789, 3457, 3467, 3469, 4567, 4679, 4789, 5689, 12347, 12379, 12457, 12479, 12569, 12589, 12689, 13457, 13469, 13567, 13679, 13789, 15679, 23459, 23567, 23689, 23789, 25679, 34589, 34679, 123457, 123479, 124567, 124679, 125789, 134789, 145679, 234589, 235679, 235789, 245789, 345679, 345689, 1234789, 1235789, 1245689, 1456789, 12356789, 23456789

MatlabEdit

Translation of: Java
queue = 1:9;

j = 1;
while j < length(queue)
    n = queue(j);
    j = j + 1;
    a = n * 10 + mod(n, 10) + 1;
    b = n * 10 + 9;
    if a <= b
        queue = [queue, a:b];    
    end
end

queue(isprime(queue))
Output:
ans =

  Columns 1 through 8

           2           3           5           7          13          17          19          23

  Columns 9 through 16

          29          37          47          59          67          79          89         127

  Columns 17 through 24

         137         139         149         157         167         179         239         257

  Columns 25 through 32

         269         347         349         359         367         379         389         457

  Columns 33 through 40

         467         479         569        1237        1249        1259        1279        1289

  Columns 41 through 48

        1367        1459        1489        1567        1579        1789        2347        2357

  Columns 49 through 56

        2389        2459        2467        2579        2689        2789        3457        3467

  Columns 57 through 64

        3469        4567        4679        4789        5689       12347       12379       12457

  Columns 65 through 72

       12479       12569       12589       12689       13457       13469       13567       13679

  Columns 73 through 80

       13789       15679       23459       23567       23689       23789       25679       34589

  Columns 81 through 88

       34679      123457      123479      124567      124679      125789      134789      145679

  Columns 89 through 96

      234589      235679      235789      245789      345679      345689     1234789     1235789

  Columns 97 through 100

     1245689     1456789    12356789    23456789

Mathematica/Wolfram LanguageEdit

ps=Sort@Select[FromDigits /@ Subsets[Range@9, {1, \[Infinity]}], PrimeQ];
Multicolumn[ps, {Automatic, 6}, Appearance -> "Horizontal"]
Output:
2	3	5	7	13	17	19	23
29	37	47	59	67	79	89	127
137	139	149	157	167	179	239	257
269	347	349	359	367	379	389	457
467	479	569	1237	1249	1259	1279	1289
1367	1459	1489	1567	1579	1789	2347	2357
2389	2459	2467	2579	2689	2789	3457	3467
3469	4567	4679	4789	5689	12347	12379	12457
12479	12569	12589	12689	13457	13469	13567	13679
13789	15679	23459	23567	23689	23789	25679	34589
34679	123457	123479	124567	124679	125789	134789	145679
234589	235679	235789	245789	345679	345689	1234789	1235789
1245689	1456789	12356789	23456789

PascalEdit

Translation of: JavaScript
{$mode Delphi}

{ Note that for the program to work properly, 
  integer variables must be at least 28-bit.
  Free Pascal Compiler uses 16-bit integers by default,
  so a directive like above is needed. }

program ascendingprimes(output);

const maxsize = 1000;

var
  queue, primes : array[1..maxsize] of integer;
  b, e, n, k, v : integer;


function isprime(n: integer): boolean;

  var
    ans : boolean;
    root, k : integer;
  begin
    if n = 2 then
      ans := true
    else if (n = 1) or (n mod 2 = 0) then
      ans := false
    else 
    begin
      root := trunc(sqrt(n));
      ans := true;
      k := 3;
      while ans and (k <= root) do
        if n mod k = 0 then
          ans := false
        else
          k := k + 2;
    end;
    isprime := ans
  end;

begin

  b := 1;
  e := 1;
  n := 0;

  for k := 1 to 9 do
  begin
    queue[e] := k;
    e := e + 1
  end;

  while b < e do
  begin
    v := queue[b];
    b := b + 1;
    if isprime(v) then
    begin
      n := n + 1;
      primes[n] := v
    end;

    for k := v mod 10 + 1 to 9 do
    begin
      queue[e] := v * 10 + k;
      e := e + 1
    end

  end;

  for k := 1 to n do
    write(primes[k], ' ');
  writeln()

end.
Output:
2 3 5 7 13 17 19 23 29 37 47 59 67 79 89 127 137 139 149 157 167 179 239 257 269 347 349 359 367 379 389 457 467 479 569 1237 1249 1259 1279 1289 1367 1459 1489 1567 1579 1789 2347 2357 2389 2459 2467 2579 2689 2789 3457 3467 3469 4567 4679 4789 5689 12347 12379 12457 12479 12569 12589 12689 13457 13469 13567 13679 13789 15679 23459 23567 23689 23789 25679 34589 34679 123457 123479 124567 124679 125789 134789 145679 234589 235679 235789 245789 345679 345689 1234789 1235789 1245689 1456789 12356789 23456789 

PerlEdit

#!/usr/bin/perl

use strict; # https://rosettacode.org/wiki/Ascending_primes
use warnings;
use ntheory qw( is_prime );

print join('', map { sprintf "%10d", $_ } sort { $a <=> $b }
  grep /./ && is_prime($_),
  glob join '', map "{$_,}", 1 .. 9) =~ s/.{50}\K/\n/gr;
Output:
         2         3         5         7        13
        17        19        23        29        37
        47        59        67        79        89
       127       137       139       149       157
       167       179       239       257       269
       347       349       359       367       379
       389       457       467       479       569
      1237      1249      1259      1279      1289
      1367      1459      1489      1567      1579
      1789      2347      2357      2389      2459
      2467      2579      2689      2789      3457
      3467      3469      4567      4679      4789
      5689     12347     12379     12457     12479
     12569     12589     12689     13457     13469
     13567     13679     13789     15679     23459
     23567     23689     23789     25679     34589
     34679    123457    123479    124567    124679
    125789    134789    145679    234589    235679
    235789    245789    345679    345689   1234789
   1235789   1245689   1456789  12356789  23456789

PhixEdit

with javascript_semantics
function ascending_primes(sequence res, atom p=0)
    for d=remainder(p,10)+1 to 9 do
        integer np = p*10+d
        if odd(d) and is_prime(np) then res &= np end if
        res = ascending_primes(res,np)
    end for
    return res
end function
 
sequence r = apply(true,sprintf,{{"%8d"},sort(ascending_primes({2}))})
printf(1,"There are %,d ascending primes:\n%s\n",{length(r),join_by(r,1,10," ")})
Output:
There are 100 ascending primes:
       2        3        5        7       13       17       19       23       29       37
      47       59       67       79       89      127      137      139      149      157
     167      179      239      257      269      347      349      359      367      379
     389      457      467      479      569     1237     1249     1259     1279     1289
    1367     1459     1489     1567     1579     1789     2347     2357     2389     2459
    2467     2579     2689     2789     3457     3467     3469     4567     4679     4789
    5689    12347    12379    12457    12479    12569    12589    12689    13457    13469
   13567    13679    13789    15679    23459    23567    23689    23789    25679    34589
   34679   123457   123479   124567   124679   125789   134789   145679   234589   235679
  235789   245789   345679   345689  1234789  1235789  1245689  1456789 12356789 23456789

powersetEdit

Using a powerset, the basic idea of which was taken from the Factor entry above, here incrementally built, does not need either recursion or a sort, same output

with javascript_semantics
function ascending_primes()
    sequence res = {}, powerset = {0}
    while length(powerset) do
        sequence next = {}
        for i=1 to length(powerset) do
            for d=remainder(powerset[i],10)+1 to 9 do
                next &= powerset[i]*10+d
            end for
        end for
        powerset = next
        res &= filter(powerset,is_prime)
    end while
    return res
end function
 
sequence r = apply(true,sprintf,{{"%8d"},ascending_primes()})
printf(1,"There are %,d ascending primes:\n%s\n",{length(r),join_by(r,1,10," ")})

By way of explanation, specifically "no sort rqd", if you pp(shorten(powerset,"entries",3)) at the end of each iteration then you get:

{1,2,3, `...`, 7,8,9, ` (9 entries)`}
{12,13,14, `...`, 78,79,89, ` (36 entries)`}
{123,124,125, `...`, 679,689,789, ` (84 entries)`}
{1234,1235,1236, `...`, 5689,5789,6789, ` (126 entries)`}
{12345,12346,12347, `...`, 45789,46789,56789, ` (126 entries)`}
{123456,123457,123458, `...`, 346789,356789,456789, ` (84 entries)`}
{1234567,1234568,1234569, `...`, 2356789,2456789,3456789, ` (36 entries)`}
{12345678,12345679,12345689, `...`, 12456789,13456789,23456789, ` (9 entries)`}
{123456789}
{}

PHPEdit

Translation of: JavaScript
<?php

function isPrime($n)
{
    if ($n == 2)
        return true;
    if ($n == 1 || $n % 2 == 0)
        return false;
    $root = intval(sqrt($n));
    for ($k = 3; $k <= $root; $k += 2)
        if ($n % $k == 0)
            return false;
    return true;    
}

$queue = [];
$primes = [];

$begin = 0;
$end = 0;

for ($k = 1; $k <= 9; $k++)
    $queue[$end++] = $k;

while ($begin < $end)
{
    $n = $queue[$begin++];

    if (isPrime($n))
        $primes[] = $n;
    for ($k = $n % 10 + 1; $k <= 9; $k++)
        $queue[$end++] = $n * 10 + $k;
}

foreach($primes as $p)
    echo "$p ";
Output:
2 3 5 7 13 17 19 23 29 37 47 59 67 79 89 127 137 139 149 157 167 179 239 257 269 347 349 359 367 379 389 457 467 479 569 1237 1249 1259 1279 1289 1367 1459 1489 1567 1579 1789 2347 2357 2389 2459 2467 2579 2689 2789 3457 3467 3469 4567 4679 4789 5689 12347 12379 12457 12479 12569 12589 12689 13457 13469 13567 13679 13789 15679 23459 23567 23689 23789 25679 34589 34679 123457 123479 124567 124679 125789 134789 145679 234589 235679 235789 245789 345679 345689 1234789 1235789 1245689 1456789 12356789 23456789 

PicatEdit

import util.

main =>
  DP = [N : S in power_set("123456789"), S != [], N = S.to_int, prime(N)].sort,
  foreach({P,I} in zip(DP,1..DP.len))
    printf("%9d%s",P,cond(I mod 10 == 0,"\n",""))
  end,
  nl,
  println(len=DP.len)
Output:
        2        3        5        7       13       17       19       23       29       37
       47       59       67       79       89      127      137      139      149      157
      167      179      239      257      269      347      349      359      367      379
      389      457      467      479      569     1237     1249     1259     1279     1289
     1367     1459     1489     1567     1579     1789     2347     2357     2389     2459
     2467     2579     2689     2789     3457     3467     3469     4567     4679     4789
     5689    12347    12379    12457    12479    12569    12589    12689    13457    13469
    13567    13679    13789    15679    23459    23567    23689    23789    25679    34589
    34679   123457   123479   124567   124679   125789   134789   145679   234589   235679
   235789   245789   345679   345689  1234789  1235789  1245689  1456789 12356789 23456789

len = 100

PicoLispEdit

(de prime? (N)
   (or
      (= N 2)
      (and
         (> N 1)
         (bit? 1 N)
         (for (D 3  T  (+ D 2))
            (T (> D (sqrt N)) T)
            (T (=0 (% N D)) NIL) ) ) ) )
(let
   (D 2
      L (1 2 2 . (4 2 4 2 4 6 2 6 .))
      Lst
      (make
         (while (>= 23456789 D)
            (and
               (prime? D)
               (apply < (chop D))
               (link D) )
            (inc 'D (++ L)) ) ) )
   (let Fmt (need 10 10)
      (while (cut 10 'Lst)
         (apply tab @ Fmt) ) ) )
Output:
         2         3         5         7        13        17        19        23        29        37
        47        59        67        79        89       127       137       139       149       157
       167       179       239       257       269       347       349       359       367       379
       389       457       467       479       569      1237      1249      1259      1279      1289
      1367      1459      1489      1567      1579      1789      2347      2357      2389      2459
      2467      2579      2689      2789      3457      3467      3469      4567      4679      4789
      5689     12347     12379     12457     12479     12569     12589     12689     13457     13469
     13567     13679     13789     15679     23459     23567     23689     23789     25679     34589
     34679    123457    123479    124567    124679    125789    134789    145679    234589    235679
    235789    245789    345679    345689   1234789   1235789   1245689   1456789  12356789  23456789

PythonEdit

Recursive solution, with a number generator and sorting of results.Edit

from sympy import isprime

def ascending(x=0):
    for y in range(x*10 + (x%10) + 1, x*10 + 10):
        yield from ascending(y)
        yield(y)

print(sorted(x for x in ascending() if isprime(x)))
Output:
[2, 3, 5, 7, 13, 17, 19, 23, 29, 37, 47, 59, 67, 79, 89, 127, 137, 139, 149, 157, 167, 179, 239, 257, 269, 347, 349, 359, 367, 379, 389, 457, 467, 479, 569, 1237, 1249, 1259, 1279, 1289, 1367, 1459, 1489, 1567, 1579, 1789, 2347, 2357, 2389, 2459, 2467, 2579, 2689, 2789, 3457, 3467, 3469, 4567, 4679, 4789, 5689, 12347, 12379, 12457, 12479, 12569, 12589, 12689, 13457, 13469, 13567, 13679, 13789, 15679, 23459, 23567, 23689, 23789, 25679, 34589, 34679, 123457, 123479, 124567, 124679, 125789, 134789, 145679, 234589, 235679, 235789, 245789, 345679, 345689, 1234789, 1235789, 1245689, 1456789, 12356789, 23456789]

Queue-based solution that does not need sorting.Edit

Translation of: Pascal
def isprime(n):
    if n == 2: return True
    if n == 1 or n % 2 == 0: return False
    root1 = int(n**0.5) + 1;
    for k in range(3, root1, 2):
        if n % k == 0: return False
    return True

queue = [k for k in range(1, 10)]
primes = []

while queue:
    n = queue.pop(0)
    if isprime(n):
        primes.append(n)
    queue.extend(n * 10 + k for k in range(n % 10 + 1, 10))

print(primes)
Output:
[2, 3, 5, 7, 13, 17, 19, 23, 29, 37, 47, 59, 67, 79, 89, 127, 137, 139, 149, 157, 167, 179, 239, 257, 269, 347, 349, 359, 367, 379, 389, 457, 467, 479, 569, 1237, 1249, 1259, 1279, 1289, 1367, 1459, 1489, 1567, 1579, 1789, 2347, 2357, 2389, 2459, 2467, 2579, 2689, 2789, 3457, 3467, 3469, 4567, 4679, 4789, 5689, 12347, 12379, 12457, 12479, 12569, 12589, 12689, 13457, 13469, 13567, 13679, 13789, 15679, 23459, 23567, 23689, 23789, 25679, 34589, 34679, 123457, 123479, 124567, 124679, 125789, 134789, 145679, 234589, 235679, 235789, 245789, 345679, 345689, 1234789, 1235789, 1245689, 1456789, 12356789, 23456789]

QuackeryEdit

powerset is defined at Power set#Quackery, and isprime is defined at Primality by trial division#Quackery.

  [ 0 swap witheach
      [ swap 10 * + ] ]                 is digits->n ( [ --> n )

  []
  ' [ 1 2 3 4 5 6 7 8 9 ] powerset
  witheach
    [ digits->n dup isprime
      iff join else drop ]
  sort echo
Output:
[ 2 3 5 7 13 17 19 23 29 37 47 59 67 79 89 127 137 139 149 157 167 179 239 257 269 347 349 359 367 379 389 457 467 479 569 1237 1249 1259 1279 1289 1367 1459 1489 1567 1579 1789 2347 2357 2389 2459 2467 2579 2689 2789 3457 3467 3469 4567 4679 4789 5689 12347 12379 12457 12479 12569 12589 12689 13457 13469 13567 13679 13789 15679 23459 23567 23689 23789 25679 34589 34679 123457 123479 124567 124679 125789 134789 145679 234589 235679 235789 245789 345679 345689 1234789 1235789 1245689 1456789 12356789 23456789 ]

RakuEdit

put (flat 2, 3, 5, 7, sort +*, gather (1..8).map: &recurse ).batch(10)».fmt("%8d").join: "\n";

sub recurse ($str) {
    .take for ($str X~ (3, 7, 9)).grep: { .is-prime && [<] .comb };
    recurse $str × 10 + $_ for $str % 10 ^.. 9;
}

printf "%.3f seconds", now - INIT now;
Output:
       2        3        5        7       13       17       19       23       29       37
      47       59       67       79       89      127      137      139      149      157
     167      179      239      257      269      347      349      359      367      379
     389      457      467      479      569     1237     1249     1259     1279     1289
    1367     1459     1489     1567     1579     1789     2347     2357     2389     2459
    2467     2579     2689     2789     3457     3467     3469     4567     4679     4789
    5689    12347    12379    12457    12479    12569    12589    12689    13457    13469
   13567    13679    13789    15679    23459    23567    23689    23789    25679    34589
   34679   123457   123479   124567   124679   125789   134789   145679   234589   235679
  235789   245789   345679   345689  1234789  1235789  1245689  1456789 12356789 23456789
0.075 seconds

RingEdit

show("ascending primes", sort(cending_primes(seq(1, 9))))

func show(title, itm)
  l = len(itm); ? "" + l + " " + title + ":"
  for i = 1 to l
    see fmt(itm[i], 9)
    if i % 5 = 0 and i < l? "" ok
  next : ? ""

func seq(b, e)
  res = []; d = e - b
  s = d / fabs(d)
  for i = b to e step s add(res, i) next
  return res

func ispr(n)
  if n < 2 return 0 ok
  if n & 1 = 0 return n = 2 ok
  if n % 3 = 0 return n = 3 ok
  l = sqrt(n)
  for f = 5 to l
    if n % f = 0 or n % (f + 2) = 0 return false ok
  next : return 1

func cending_primes(digs)
  cand = [0]
  pr = []
  for i in digs
    lcand = cand
    for j in lcand
      v = j * 10 + i
      if ispr(v) add(pr, v) ok
      add(cand, v)
    next
  next
  return pr

func fmt(x, l)
  res = "          " + x
  return right(res, l)
Output:
100 ascending primes:
        2        3        5        7       13
       17       19       23       29       37
       47       59       67       79       89
      127      137      139      149      157
      167      179      239      257      269
      347      349      359      367      379
      389      457      467      479      569
     1237     1249     1259     1279     1289
     1367     1459     1489     1567     1579
     1789     2347     2357     2389     2459
     2467     2579     2689     2789     3457
     3467     3469     4567     4679     4789
     5689    12347    12379    12457    12479
    12569    12589    12689    13457    13469
    13567    13679    13789    15679    23459
    23567    23689    23789    25679    34589
    34679   123457   123479   124567   124679
   125789   134789   145679   234589   235679
   235789   245789   345679   345689  1234789
  1235789  1245689  1456789 12356789 23456789

RubyEdit

require 'prime'

digits = [9,8,7,6,5,4,3,2,1]
res = 1.upto(digits.size).flat_map do |n|
   digits.combination(n).filter_map do |set|
      candidate = set.join.to_i
      candidate if candidate.prime?
   end.reverse
 end

 puts res.join(",")
Output:
2,3,5,7,13,17,19,23,29,37,47,59,67,79,89,127,137,139,149,157,167,179,239,257,269,347,349,359,367,379,389,457,467,479,569,1237,1249,1259,1279,1289,1367,1459,1489,1567,1579,1789,2347,2357,2389,2459,2467,2579,2689,2789,3457,3467,3469,4567,4679,4789,5689,12347,12379,12457,12479,12569,12589,12689,13457,13469,13567,13679,13789,15679,23459,23567,23689,23789,25679,34589,34679,123457,123479,124567,124679,125789,134789,145679,234589,235679,235789,245789,345679,345689,1234789,1235789,1245689,1456789,12356789,23456789

SidefEdit

func primes_with_ascending_digits(base = 10) {

    var list = []
    var digits = @(1..^base -> flip)

    var end_digits = digits.grep { .is_coprime(base) }
    list << digits.grep { .is_prime && !.is_coprime(base) }...

    for k in (0 .. digits.end) {
        digits.combinations(k, {|*a|
            var v = a.digits2num(base)
            end_digits.each {|d|
                var n = (v*base + d)
                next if ((n >= base) && (a[0] >= d))
                list << n if (n.is_prime)
            }
        })
    }

    list.sort
}

var arr = primes_with_ascending_digits()

say "There are #{arr.len} ascending primes.\n"

arr.each_slice(10, {|*a|
    say a.map { '%8s' % _ }.join(' ')
})
Output:
There are 100 ascending primes.

       2        3        5        7       13       17       19       23       29       37
      47       59       67       79       89      127      137      139      149      157
     167      179      239      257      269      347      349      359      367      379
     389      457      467      479      569     1237     1249     1259     1279     1289
    1367     1459     1489     1567     1579     1789     2347     2357     2389     2459
    2467     2579     2689     2789     3457     3467     3469     4567     4679     4789
    5689    12347    12379    12457    12479    12569    12589    12689    13457    13469
   13567    13679    13789    15679    23459    23567    23689    23789    25679    34589
   34679   123457   123479   124567   124679   125789   134789   145679   234589   235679
  235789   245789   345679   345689  1234789  1235789  1245689  1456789 12356789 23456789

Visual Basic .NETEdit

Module AscendingPrimes

    Function isPrime(n As Integer)
        n = Math.Abs(n)
        If n = 2 Then
            Return True
        End If
        If n = 1 Or n Mod 2 = 0 Then
            Return False
        End If
        Dim root As Integer = Math.Sqrt(n)
        For k = 3 To root Step 2
            If n Mod k = 0 Then
                Return False
            End If
        Next
        Return True
    End Function


    Sub Main()

        Dim queue As Queue(Of Integer) = New Queue(Of Integer)
        Dim primes As List(Of Integer) = New List(Of Integer)

        For k = 1 To 9
            queue.Enqueue(k)
        Next

        While queue.Count > 0
            Dim n As Integer = queue.Dequeue()
            If (isPrime(n)) Then
                primes.Add(n)
            End If
            For k = n Mod 10 + 1 To 9
                queue.Enqueue(n * 10 + k)
            Next
        End While

        For Each p As Integer In primes
            Console.Write(p)
            Console.Write(" ")
        Next
        Console.WriteLine()

    End Sub

End Module
Output:
2 3 5 7 13 17 19 23 29 37 47 59 67 79 89 127 137 139 149 157 167 179 239 257 269 347 349 359 367 379 389 457 467 479 569 1237 1249 1259 1279 1289 1367 1459 1489 1567 1579 1789 2347 2357 2389 2459 2467 2579 2689 2789 3457 3467 3469 4567 4679 4789 5689 12347 12379 12457 12479 12569 12589 12689 13457 13469 13567 13679 13789 15679 23459 23567 23689 23789 25679 34589 34679 123457 123479 124567 124679 125789 134789 145679 234589 235679 235789 245789 345679 345689 1234789 1235789 1245689 1456789 12356789 23456789

V (Vlang)Edit

Translation of: Go
fn is_prime(n int) bool {
    if n < 2 {
        return false
    } else if n%2 == 0 {
        return n == 2
    } else if n%3 == 0 {
        return n == 3
    } else {
        mut d := 5
        for d*d <= n {
            if n%d == 0 {
                return false
            }
            d += 2
            if n%d == 0 {
                return false
            }
            d += 4
        }
        return true
    }
}
fn generate(first int, cand int, digits int, mut asc map[int]bool) {
    if digits == 0 {
        if is_prime(cand) {
            asc[cand] = true
        }
        return
    }
    for i in first..10 {
        next := cand*10 + i
        generate(i+1, next, digits-1, mut asc)
    }
}
 
fn main() {
    mut asc_primes_set := map[int]bool{} // avoids duplicates

    for digits in 1..10 {
        generate(1, 0, digits, mut asc_primes_set)
    }
    le := asc_primes_set.keys().len
    mut asc_primes := []int{len: le}
    mut i := 0
    for k,_ in asc_primes_set {
        asc_primes[i] = k
        i++
    }
    asc_primes.sort()
    println("There are $le ascending primes, namely:")
    for q in 0..le {
        print("${asc_primes[q]:8} ")
        if (q+1)%10 == 0 {
            println('')
        }
    }
}
Output:
There are 100 ascending primes, namely:
       2        3        5        7       13       17       19       23       29       37 
      47       59       67       79       89      127      137      139      149      157 
     167      179      239      257      269      347      349      359      367      379 
     389      457      467      479      569     1237     1249     1259     1279     1289 
    1367     1459     1489     1567     1579     1789     2347     2357     2389     2459 
    2467     2579     2689     2789     3457     3467     3469     4567     4679     4789 
    5689    12347    12379    12457    12479    12569    12589    12689    13457    13469 
   13567    13679    13789    15679    23459    23567    23689    23789    25679    34589 
   34679   123457   123479   124567   124679   125789   134789   145679   234589   235679 
  235789   245789   345679   345689  1234789  1235789  1245689  1456789 12356789 23456789 

WrenEdit

Library: Wren-math
Library: Wren-seq
Library: Wren-fmt

Version 1 (Sieve)Edit

Although they use a lot of memory, sieves usually produce good results in Wren and here we only need to sieve for primes up to 3456789 as there are just 9 possible candidates with 8 digits and 1 possible candidate with 9 digits which we can test for primality individually. The following runs in around 0.43 seconds.

import "./math" for Int
import "./seq"  for Lst
import "./fmt"  for Fmt

var isAscending = Fn.new { |n|
    if (n < 10) return true
    var digits = Int.digits(n)
    for (i in 1...digits.count) {
        if (digits[i] <= digits[i-1]) return false
    }
    return true
}

var higherPrimes = []
var candidates = [
    12345678, 12345679, 12345689, 12345789, 12346789,
    12356789, 12456789, 13456789, 23456789, 123456789 
]
for (cand in candidates) if (Int.isPrime(cand)) higherPrimes.add(cand)

var primes = Int.primeSieve(3456789)
var ascPrimes = []
for (p in primes) if (isAscending.call(p)) ascPrimes.add(p)
ascPrimes.addAll(higherPrimes)
System.print("There are %(ascPrimes.count) ascending primes, namely:")
for (chunk in Lst.chunks(ascPrimes, 10)) Fmt.print("$8d", chunk)
Output:
There are 100 ascending primes, namely:
       2        3        5        7       13       17       19       23       29       37
      47       59       67       79       89      127      137      139      149      157
     167      179      239      257      269      347      349      359      367      379
     389      457      467      479      569     1237     1249     1259     1279     1289
    1367     1459     1489     1567     1579     1789     2347     2357     2389     2459
    2467     2579     2689     2789     3457     3467     3469     4567     4679     4789
    5689    12347    12379    12457    12479    12569    12589    12689    13457    13469
   13567    13679    13789    15679    23459    23567    23689    23789    25679    34589
   34679   123457   123479   124567   124679   125789   134789   145679   234589   235679
  235789   245789   345679   345689  1234789  1235789  1245689  1456789 12356789 23456789

Version 2 (Generator)Edit

Library: Wren-set

Here we generate all possible positive integers with ascending non-zero digits and filter out those that are prime.

Much quicker than the 'sieve' approach at 0.013 seconds. I also tried using a powerset but that was slightly slower at 0.015 seconds.

import "./set" for Set
import "./math" for Int
import "./seq" for Lst
import "./fmt" for Fmt

var ascPrimes = Set.new() // avoids duplicates

var generate  // recursive function
generate = Fn.new { |first, cand, digits|
    if (digits == 0) {
        if (Int.isPrime(cand)) ascPrimes.add(cand)
        return
    }
    var i = first
    while (i <= 9) {
        var next = cand * 10 + i
        generate.call(i + 1, next, digits - 1)
        i = i + 1
    }
}

for (digits in 1..9) generate.call(1, 0, digits)
ascPrimes = ascPrimes.toList
ascPrimes.sort()
System.print("There are %(ascPrimes.count) ascending primes, namely:")
for (chunk in Lst.chunks(ascPrimes, 10)) Fmt.print("$8s", chunk)
Output:
Same as before.

XPL0Edit

Brute force solution: 4.3 seconds on Pi4.

func IsPrime(N);        \Return 'true' if N is prime
int  N, I;
[if N <= 2 then return N = 2;
if (N&1) = 0 then \even >2\ return false;
for I:= 3 to sqrt(N) do
    [if rem(N/I) = 0 then return false;
    I:= I+1;
    ];
return true;
];

func Ascending(N);      \Return 'true' if digits are ascending
int  N, D;
[N:= N/10;
D:= rem(0);
while N do
    [N:= N/10;
    if rem(0) >= D then return false;
    D:= rem(0);
    ];
return true;
];

int Cnt, N;
[Cnt:= 0;
Format(9, 0);
for N:= 2 to 123_456_789 do
    if Ascending(N) then
        if IsPrime(N) then
            [RlOut(0, float(N));
            Cnt:= Cnt+1;
            if rem(Cnt/10) = 0 then CrLf(0);
            ];
]
Output:
        2        3        5        7       13       17       19       23       29       37
       47       59       67       79       89      127      137      139      149      157
      167      179      239      257      269      347      349      359      367      379
      389      457      467      479      569     1237     1249     1259     1279     1289
     1367     1459     1489     1567     1579     1789     2347     2357     2389     2459
     2467     2579     2689     2789     3457     3467     3469     4567     4679     4789
     5689    12347    12379    12457    12479    12569    12589    12689    13457    13469
    13567    13679    13789    15679    23459    23567    23689    23789    25679    34589
    34679   123457   123479   124567   124679   125789   134789   145679   234589   235679
   235789   245789   345679   345689  1234789  1235789  1245689  1456789 12356789 23456789

powersetEdit

Aaah! Power set, from Factor. Runs in less than 1 millisecond. A better way of measuring duration than using Linux's time utility gave a more credible 35 milliseconds.

include xpllib;         \provides IsPrime and Sort

int I, N, Mask, Digit, A(512), Cnt;
[for I:= 0 to 511 do
    [N:= 0;  Mask:= I;  Digit:= 1;
    while Mask do
        [if Mask&1 then 
            N:= N*10 + Digit;
        Mask:= Mask>>1;
        Digit:= Digit+1;
        ];
    A(I):= N;
    ];
Sort(A, 512);
Cnt:= 0;
Format(9, 0);
for I:= 1 to 511 do     \skip empty set
    [N:= A(I);
    if IsPrime(N) then
        [RlOut(0, float(N));
        Cnt:= Cnt+1;
        if rem(Cnt/10) = 0 then CrLf(0);
        ];
    ];
]
Output:
Same as before.