I'm working on modernizing Rosetta Code's infrastructure. Starting with communications. Please accept this time-limited open invite to RC's Slack.. --Michael Mol (talk) 20:59, 30 May 2020 (UTC)

# Numbers with prime digits whose sum is 13

Numbers with prime digits whose sum is 13 is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Find all the positive integers whose decimal digits are all primes and sum to   13.

## 11l

Translation of: C
F primeDigitsSum13(=n)
V sum = 0
L n > 0
V r = n % 10
I r !C (2, 3, 5, 7)
R 0B
n I/= 10
sum += r
R sum == 13

V c = 0
L(i) 1..999999
I primeDigitsSum13(i)
print(‘#6’.format(i), end' ‘ ’)
I ++c == 11
c = 0
print()
print()
Output:
337    355    373    535    553    733   2227   2272   2335   2353   2533
2722   3235   3253   3325   3352   3523   3532   5233   5323   5332   7222
22225  22252  22333  22522  23233  23323  23332  25222  32233  32323  32332
33223  33232  33322  52222 222223 222232 222322 223222 232222 322222

## Action!

DEFINE PRIMECOUNT="4"
BYTE ARRAY primedigits(PRIMECOUNT)=[2 3 5 7]

PROC PrintNum(BYTE ARRAY code BYTE count)
BYTE i,c

FOR i=0 TO count-1
DO
c=code(i)
Put(primedigits(c)+'0)
OD
RETURN

BYTE FUNC Sum(BYTE ARRAY code BYTE count)
BYTE i,res,c

res=0
FOR i=0 TO count-1
DO
c=code(i)
res==+primedigits(c)
OD
RETURN (res)

PROC Init(BYTE ARRAY code BYTE count)
Zero(code,count)
RETURN

BYTE FUNC Next(BYTE ARRAY code BYTE count)
INT pos,c

pos=count-1
DO
c=code(pos)+1
IF c<PRIMECOUNT THEN
code(pos)=c
RETURN (1)
FI
code(pos)=0
pos==-1
IF pos<0 THEN
RETURN (0)
FI
OD
RETURN (0)

PROC Main()
DEFINE MAXDIG="6"
BYTE ARRAY digits(MAXDIG)
BYTE count,pos

count=1 pos=0
Init(count)
DO
IF Sum(digits,count)=13 THEN
IF pos+count>=38 THEN
pos=0 PutE()
FI
PrintNum(digits,count) Put(32)
pos==+count+1
FI
IF Next(digits,count)=0 THEN
count==+1
IF count>MAXDIG THEN
EXIT
FI
Init(count)
FI
OD
RETURN
Output:
337 355 373 535 553 733 2227 2272 2335 2353 2533 2722 3235 3253 3325 3352 3523 3532 5233 5323 5332 7222
22225 22252 22333 22522 23233 23323 23332 25222 32233 32323 32332 33223 33232 33322 52222 222223 222232
222322 223222 232222 322222

## ALGOL 68

Based on the Algol W sample.

BEGIN
# find numbers whose digits are prime and whose digit sum is 13 #
# as noted by the Wren sample, the digits can only be 2, 3, 5, 7 #
# and there can only be 3, 4, 5 or 6 digits #
[]INT possible digits = []INT( 0, 2, 3, 5, 7 )[ AT 0 ];
INT number count := 0;
INT zero = ABS "0"; # integer value of the character "0" #
print( ( newline ) );
FOR d1 index FROM 0 TO UPB possible digits DO
INT d1 = possible digits[ d1 index ];
CHAR c1 = IF d1 /= 0 THEN REPR ( d1 + zero ) ELSE " " FI;
FOR d2 index FROM 0 TO UPB possible digits DO
INT d2 = possible digits[ d2 index ];
IF d2 /= 0 OR d1 = 0 THEN
CHAR c2 = IF ( d1 + d2 ) /= 0 THEN REPR ( d2 + zero ) ELSE " " FI;
FOR d3 index FROM 0 TO UPB possible digits DO
INT d3 = possible digits[ d3 index ];
IF d3 /= 0 OR ( d1 + d2 ) = 0 THEN
CHAR c3 = IF ( d1 + d2 + d3 ) /= 0 THEN REPR ( d3 + zero ) ELSE " " FI;
FOR d4 index FROM 1 TO UPB possible digits DO
INT d4 = possible digits[ d4 index ];
CHAR c4 = REPR ( d4 + zero );
FOR d5 index FROM 1 TO UPB possible digits DO
INT d5 = possible digits[ d5 index ];
CHAR c5 = REPR ( d5 + zero );
FOR d6 index FROM 1 TO UPB possible digits DO
INT d6 = possible digits[ d6 index ];
IF ( d1 + d2 + d3 + d4 + d5 + d6 ) = 13 THEN
# found a number whose prime digits sum to 13 #
CHAR c6 = REPR ( d6 + zero );
print( ( " ", c1, c2, c3, c4, c5, c6 ) );
number count := number count + 1;
IF ( number count +:= 1 ) MOD 12 = 0 THEN print( ( newline ) ) FI
FI
OD # d6 #
OD # d5 #
OD # d4 #
FI
OD # d3 #
FI
OD # d2 #
OD # d1 #
END
Output:
337    355    373    535    553    733   2227   2272   2335   2353   2533   2722
3235   3253   3325   3352   3523   3532   5233   5323   5332   7222  22225  22252
22333  22522  23233  23323  23332  25222  32233  32323  32332  33223  33232  33322
52222 222223 222232 222322 223222 232222 322222

## ALGOL W

Uses the observations about the digits and numbers in the Wren solution to generate the sequence.

begin
% find numbers whose digits are prime and whose digit sum is 13  %
% as noted by the Wren sample, the digits can only be 2, 3, 5, 7 %
% and there can only be 3, 4, 5 or 6 digits  %
integer numberCount;
numberCount := 0;
write();
for d1 := 0, 2, 3, 5, 7 do begin
for d2 := 0, 2, 3, 5, 7 do begin
if d2 not = 0 or d1 = 0 then begin
for d3 := 0, 2, 3, 5, 7 do begin
if d3 not = 0 or ( d1 = 0 and d2 = 0 ) then begin
for d4 := 2, 3, 5, 7 do begin
for d5 := 2, 3, 5, 7 do begin
for d6 := 2, 3, 5, 7 do begin
integer sum;
sum := d1 + d2 + d3 + d4 + d5 + d6;
if sum = 13 then begin
% found a number whose prime digits sum to 13 %
integer n;
n := 0;
for d := d1, d2, d3, d4, d5, d6 do n := ( n * 10 ) + d;
writeon( i_w := 6, s_w := 1, n );
numberCount := numberCount + 1;
if numberCount rem 12 = 0 then write()
end if_sum_eq_13
end for_d6
end for_d5
end for_d4
end if_d3_ne_0_or_d1_eq_0_and_d2_e_0
end for_d3
end if_d2_ne_0_or_d1_eq_0
end for_d2
end for_d1
end.
Output:
337    355    373    535    553    733   2227   2272   2335   2353   2533   2722
3235   3253   3325   3352   3523   3532   5233   5323   5332   7222  22225  22252
22333  22522  23233  23323  23332  25222  32233  32323  32332  33223  33232  33322
52222 222223 222232 222322 223222 232222 322222

## Arturo

pDigits: [2 3 5 7]

lst: map pDigits 'd -> @[d]
result: new []

while [0 <> size lst][
nextList: new []
loop lst 'digitSeq [
currSum: sum digitSeq
loop pDigits 'n [
newSum: currSum + n
newDigitSeq: digitSeq ++ n
case [newSum]
when? [<13] -> 'nextList ++ @[newDigitSeq]
when? [=13] -> 'result ++ @[to :integer join to [:string] newDigitSeq]
else -> break
]
]
lst: new nextList
]

loop split.every: 10 result 'a ->
print map a => [pad to :string & 6]
Output:
337    355    373    535    553    733   2227   2272   2335   2353
2533   2722   3235   3253   3325   3352   3523   3532   5233   5323
5332   7222  22225  22252  22333  22522  23233  23323  23332  25222
32233  32323  32332  33223  33232  33322  52222 222223 222232 222322
223222 232222 322222

## AWK

# syntax: GAWK -f NUMBERS_WITH_PRIME_DIGITS_WHOSE_SUM_IS_13.AWK
BEGIN {
for (i=1; i<=1000000; i++) {
if (prime_digits_sum13(i)) {
printf("%6d ",i)
if (++count % 10 == 0) {
printf("\n")
}
}
}
printf("\n")
exit(0)
}
function prime_digits_sum13(n, r,sum) {
while (n > 0) {
r = int(n % 10)
switch (r) {
case 2:
case 3:
case 5:
case 7:
break
default:
return(0)
}
n = int(n / 10)
sum += r
}
return(sum == 13)
}

Output:
337    355    373    535    553    733   2227   2272   2335   2353
2533   2722   3235   3253   3325   3352   3523   3532   5233   5323
5332   7222  22225  22252  22333  22522  23233  23323  23332  25222
32233  32323  32332  33223  33232  33322  52222 222223 222232 222322
223222 232222 322222

## C

Brute force

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

bool primeDigitsSum13(int n) {
int sum = 0;
while (n > 0) {
int r = n % 10;
switch (r) {
case 2:
case 3:
case 5:
case 7:
break;
default:
return false;
}
n /= 10;
sum += r;
}
return sum == 13;
}

int main() {
int i, c;

// using 2 for all digits, 6 digits is the max prior to over-shooting 13
c = 0;
for (i = 1; i < 1000000; i++) {
if (primeDigitsSum13(i)) {
printf("%6d ", i);
if (c++ == 10) {
c = 0;
printf("\n");
}
}
}
printf("\n");

return 0;
}
Output:
337    355    373    535    553    733   2227   2272   2335   2353   2533
2722   3235   3253   3325   3352   3523   3532   5233   5323   5332   7222
22225  22252  22333  22522  23233  23323  23332  25222  32233  32323  32332
33223  33232  33322  52222 222223 222232 222322 223222 232222 322222

## C#

Translation of: Phix

Same recursive method.

using System;
using static System.Console;
using LI = System.Collections.Generic.SortedSet<int>;

class Program {

static LI unl(LI res, LI set, int lft, int mul = 1, int vlu = 0) {
else if (lft > 0) foreach (int itm in set)
res = unl(res, set, lft - itm, mul * 10, vlu + itm * mul);
return res; }

static void Main(string[] args) { WriteLine(string.Join(" ",
unl(new LI {}, new LI { 2, 3, 5, 7 }, 13))); }
}
Output:
337 355 373 535 553 733 2227 2272 2335 2353 2533 2722 3235 3253 3325 3352 3523 3532 5233 5323 5332 7222 22225 22252 22333 22522 23233 23323 23332 25222 32233 32323 32332 33223 33232 33322 52222 222223 222232 222322 223222 232222 322222

### Alternate

Based in Nigel Galloway's suggestion from the discussion page.

class Program {

static void Main(string[] args) { int[] lst; int sum;
var w = new System.Collections.Generic.List<(int digs, int sum)> {};
foreach (int x in lst = new int[] { 2, 3, 5, 7 } ) w.Add((x, x));
while (w.Count > 0) { var i = w[0]; w.RemoveAt(0);
foreach (var j in lst) if ((sum = i.sum + j) == 13)
System.Console.Write ("{0}{1} ", i.digs, j);
else if (sum < 12)
w.Add((i.digs * 10 + j, sum)); } }
}

Same output.

## C++

Translation of: C#
(the alternate version)
#include <cstdio>
#include <vector>
#include <bits/stdc++.h>

using namespace std;

int main() {
vector<tuple<int, int>> w; int lst[4] = { 2, 3, 5, 7 }, sum;
for (int x : lst) w.push_back({x, x});
while (w.size() > 0) { auto i = w[0]; w.erase(w.begin());
for (int x : lst) if ((sum = get<1>(i) + x) == 13)
printf("%d%d ", get<0>(i), x);
else if (sum < 12) w.push_back({get<0>(i) * 10 + x, sum}); }
return 0; }

Same output as C#.

## D

Translation of: C
import std.stdio;

bool primeDigitsSum13(int n) {
int sum = 0;
while (n > 0) {
int r = n % 10;
switch (r) {
case 2,3,5,7:
break;
default:
return false;
}
n /= 10;
sum += r;
}
return sum == 13;
}

void main() {
// using 2 for all digits, 6 digits is the max prior to over-shooting 13
int c = 0;
for (int i = 1; i < 1_000_000; i++) {
if (primeDigitsSum13(i)) {
writef("%6d ", i);
if (c++ == 10) {
c = 0;
writeln;
}
}
}
writeln;
}
Output:
337    355    373    535    553    733   2227   2272   2335   2353   2533
2722   3235   3253   3325   3352   3523   3532   5233   5323   5332   7222
22225  22252  22333  22522  23233  23323  23332  25222  32233  32323  32332
33223  33232  33322  52222 222223 222232 222322 223222 232222 322222

## F#

// prime digits whose sum is 13. Nigel Galloway: October 21st., 2020
let rec fN g=let g=[for n in [2;3;5;7] do for g in g->n::g]|>List.groupBy(fun n->match List.sum n with 13->'n' |n when n<12->'g' |_->'x')|>Map.ofSeq
[yield! (if g.ContainsKey 'n' then g.['n'] else []); yield! (if g.ContainsKey 'g' then fN g.['g'] else [])]
fN [[]] |> Seq.iter(fun n->n|>List.iter(printf "%d");printf " ");printfn ""

Output:
337 355 373 535 553 733 2227 2272 2335 2353 2533 2722 3235 3253 3325 3352 3523 3532 5233 5323 5332 7222 22225 22252 22333 22522 23233 23323 23332 25222 32233 32323 32332 33223 33232 33322 52222 222223 222232 222322 223222 232222 322222

## Factor

### Filtering selections

Generate all selections of the prime digits in the only possible lengths whose sum can be 13, then filter for sums that equal 13.

USING: formatting io kernel math math.combinatorics
math.functions math.ranges sequences sequences.extras ;

: digits>number ( seq -- n ) reverse 0 [ 10^ * + ] reduce-index ;

"Numbers whose digits are prime and sum to 13:" print
{ 2 3 5 7 } 3 6 [a,b] [ selections [ sum 13 = ] filter ] with
map-concat [ digits>number ] map "%[%d, %]\n" printf
Output:
Numbers whose digits are prime and sum to 13:
{ 337, 355, 373, 535, 553, 733, 2227, 2272, 2335, 2353, 2533, 2722, 3235, 3253, 3325, 3352, 3523, 3532, 5233, 5323, 5332, 7222, 22225, 22252, 22333, 22522, 23233, 23323, 23332, 25222, 32233, 32323, 32332, 33223, 33232, 33322, 52222, 222223, 222232, 222322, 223222, 232222, 322222 }

### F# translation

The following is based on Nigel Galloway's algorithm as described here on the talk page. It's about 10x faster than the previous method.

USING: io kernel math prettyprint sequences sequences.extras ;

{ } { { 2 } { 3 } { 5 } { 7 } } [
{ 2 3 5 7 } [ suffix ] cartesian-map concat
[ sum 13 = ] partition [ append ] dip [ sum 11 > ] reject
] until-empty [ bl ] [ [ pprint ] each ] interleave nl
Output:
337 355 373 535 553 733 2227 2272 2335 2353 2533 2722 3235 3253 3325 3352 3523 3532 5233 5323 5332 7222 22225 22252 22333 22522 23233 23323 23332 25222 32233 32323 32332 33223 33232 33322 52222 222223 222232 222322 223222 232222 322222

## FreeBASIC

Ho hum. Another prime digits task.

function digit_is_prime( n as integer ) as boolean
select case n
case 2,3,5,7
return true
case else
return false
end select
end function

function all_digits_prime( n as uinteger ) as boolean
dim as string sn = str(n)
for i as uinteger = 1 to len(sn)
if not digit_is_prime( val(mid(sn,i,1)) ) then return false
next i
return true
end function

function digit_sum_13( n as uinteger ) as boolean
dim as string sn = str(n)
dim as integer k = 0
for i as uinteger = 1 to len(sn)
k = k + val(mid(sn,i,1))
if k>13 then return false
next i
if k<>13 then return false else return true
end function

for i as uinteger = 1 to 322222
if all_digits_prime(i) andalso digit_sum_13(i) then print i,
next i
Output:
337           355           373           535           553           733
2227          2272          2335          2353          2533          2722
3235          3253          3325          3352          3523          3532
5233          5323          5332          7222          22225         22252
22333         22522         23233         23323         23332         25222
32233         32323         32332         33223         33232         33322
52222         222223        222232        222322        223222        232222
322222

## Go

Reuses code from some other tasks.

package main

import (
"fmt"
"sort"
"strconv"
)

func combrep(n int, lst []byte) [][]byte {
if n == 0 {
return [][]byte{nil}
}
if len(lst) == 0 {
return nil
}
r := combrep(n, lst[1:])
for _, x := range combrep(n-1, lst) {
r = append(r, append(x, lst[0]))
}
return r
}

func shouldSwap(s []byte, start, curr int) bool {
for i := start; i < curr; i++ {
if s[i] == s[curr] {
return false
}
}
return true
}

func findPerms(s []byte, index, n int, res *[]string) {
if index >= n {
*res = append(*res, string(s))
return
}
for i := index; i < n; i++ {
check := shouldSwap(s, index, i)
if check {
s[index], s[i] = s[i], s[index]
findPerms(s, index+1, n, res)
s[index], s[i] = s[i], s[index]
}
}
}

func main() {
primes := []byte{2, 3, 5, 7}
var res []string
for n := 3; n <= 6; n++ {
reps := combrep(n, primes)
for _, rep := range reps {
sum := byte(0)
for _, r := range rep {
sum += r
}
if sum == 13 {
var perms []string
for i := 0; i < len(rep); i++ {
rep[i] += 48
}
findPerms(rep, 0, len(rep), &perms)
res = append(res, perms...)
}
}
}
res2 := make([]int, len(res))
for i, r := range res {
res2[i], _ = strconv.Atoi(r)
}
sort.Ints(res2)
fmt.Println("Those numbers whose digits are all prime and sum to 13 are:")
fmt.Println(res2)
}
Output:
Those numbers whose digits are all prime and sum to 13 are:
[337 355 373 535 553 733 2227 2272 2335 2353 2533 2722 3235 3253 3325 3352 3523 3532 5233 5323 5332 7222 22225 22252 22333 22522 23233 23323 23332 25222 32233 32323 32332 33223 33232 33322 52222 222223 222232 222322 223222 232222 322222]

### only counting

See Julia [1]

package main

import (
"fmt"
)
var
Primes = []byte{2, 3, 5, 7};
var
gblCount = 0;
var
PrimesIdx = []byte{0, 1, 2, 3};

func combrep(n int, lst []byte) [][]byte {
if n == 0 {
return [][]byte{nil}
}
if len(lst) == 0 {
return nil
}
r := combrep(n, lst[1:])
for _, x := range combrep(n-1, lst) {
r = append(r, append(x, lst[0]))
}
return r
}

func Count(rep []byte)int {
var PrimCount [4]int
for i := 0; i < len(PrimCount); i++ {
PrimCount[i] = 0;
}
//get the count of every item
for i := 0; i < len(rep); i++ {
PrimCount[rep[i]]++
}
var numfac int = len(rep)

var numerator,denominator[]int

for i := 1; i <= len(rep); i++ {
numerator = append(numerator,i) // factors 1,2,3,4. n
denominator = append(denominator,1)
}
numfac = 0; //idx in denominator
for i := 0; i < len(PrimCount); i++ {
denfac := 1;
for j := 0; j < PrimCount[i]; j++ {
denominator[numfac] = denfac
denfac++
numfac++
}
}
//calculate permutations with identical items
numfac = 1;
for i := 0; i < len(numerator); i++ {
numfac = (numfac * numerator[i])/denominator[i]
}
return numfac
}

func main() {
for mySum := 2; mySum <= 103;mySum++ {
gblCount = 0;
//check for prime
for i := 2; i*i <= mySum;i++{
if mySum%i == 0 {
gblCount=1;
break
}
}
if gblCount != 0 {
continue
}

for n := 1; n <= mySum / 2 ; n++ {
reps := combrep(n, PrimesIdx)
for _, rep := range reps {
sum := byte(0)
for _, r := range rep {
sum += Primes[r]
}
if sum == byte(mySum) {
gblCount+=Count(rep);
}
}
}
fmt.Println("The count of numbers whose digits are all prime and sum to",mySum,"is",gblCount)
}
}
Output:
The count of numbers whose digits are all prime and sum to 2 is 1
The count of numbers whose digits are all prime and sum to 3 is 1
The count of numbers whose digits are all prime and sum to 5 is 3
The count of numbers whose digits are all prime and sum to 7 is 6
The count of numbers whose digits are all prime and sum to 11 is 19
The count of numbers whose digits are all prime and sum to 13 is 43
The count of numbers whose digits are all prime and sum to 17 is 221
The count of numbers whose digits are all prime and sum to 19 is 468
The count of numbers whose digits are all prime and sum to 23 is 2098
The count of numbers whose digits are all prime and sum to 29 is 21049
The count of numbers whose digits are all prime and sum to 31 is 45148
The count of numbers whose digits are all prime and sum to 37 is 446635
The count of numbers whose digits are all prime and sum to 41 is 2061697
The count of numbers whose digits are all prime and sum to 43 is 4427752
The count of numbers whose digits are all prime and sum to 47 is 20424241
The count of numbers whose digits are all prime and sum to 53 is 202405001
The count of numbers whose digits are all prime and sum to 59 is 2005642061
The count of numbers whose digits are all prime and sum to 61 is 4307930784
The count of numbers whose digits are all prime and sum to 67 is 42688517778
The count of numbers whose digits are all prime and sum to 71 is 196942068394
The count of numbers whose digits are all prime and sum to 73 is 423011795680
The count of numbers whose digits are all prime and sum to 79 is 4191737820642
The count of numbers whose digits are all prime and sum to 83 is 19338456915087
The count of numbers whose digits are all prime and sum to 89 is 191629965405641
The count of numbers whose digits are all prime and sum to 97 is 4078672831913824
The count of numbers whose digits are all prime and sum to 101 is 18816835854129198
The count of numbers whose digits are all prime and sum to 103 is 40416663565084464

real  0m4,489s
user  0m5,584s
sys 0m0,188s

As an unfold, in the recursive pattern described by Nigel Galloway on the Talk page.

import Data.List.Split (chunksOf)
import Data.List (intercalate, transpose, unfoldr)
import Text.Printf

primeDigitsNumsSummingToN :: Int -> [Int]
primeDigitsNumsSummingToN n = concat \$ unfoldr go (return <\$> primeDigits)
where
primeDigits = [2, 3, 5, 7]

go :: [[Int]] -> Maybe ([Int], [[Int]])
go xs
| null xs = Nothing
| otherwise = Just (nextLength xs)

nextLength :: [[Int]] -> ([Int], [[Int]])
nextLength xs =
let harvest nv =
[ unDigits \$ fst nv
| n == snd nv ]
prune nv =
[ fst nv
| pred n > snd nv ]
in ((,) . concatMap harvest <*> concatMap prune)
(((,) <*> sum) <\$> ((<\$> xs) . (<>) . return =<< primeDigits))

--------------------------- TEST -------------------------
main :: IO ()
main = do
let n = 13
xs = primeDigitsNumsSummingToN n
mapM_
putStrLn
[ concat
[ (show . length) xs
, " numbers with prime digits summing to "
, show n
, ":\n"
]
, table " " \$ chunksOf 10 (show <\$> xs)
]

table :: String -> [[String]] -> String
table gap rows =
let ic = intercalate
ws = maximum . fmap length <\$> transpose rows
pw = printf . flip ic ["%", "s"] . show
in unlines \$ ic gap . zipWith pw ws <\$> rows

unDigits :: [Int] -> Int
unDigits = foldl ((+) . (10 *)) 0
Output:
43 numbers with prime digits summing to 13:

337    355    373   535   553   733  2227   2272   2335   2353
2533   2722   3235  3253  3325  3352  3523   3532   5233   5323
5332   7222  22225 22252 22333 22522 23233  23323  23332  25222
32233  32323  32332 33223 33232 33322 52222 222223 222232 222322
223222 232222 322222

## Java

Translation of: Kotlin
public class PrimeDigits {
private static boolean primeDigitsSum13(int n) {
int sum = 0;
while (n > 0) {
int r = n % 10;
if (r != 2 && r != 3 && r != 5 && r != 7) {
return false;
}
n /= 10;
sum += r;
}
return sum == 13;
}

public static void main(String[] args) {
// using 2 for all digits, 6 digits is the max prior to over-shooting 13
int c = 0;
for (int i = 1; i < 1_000_000; i++) {
if (primeDigitsSum13(i)) {
System.out.printf("%6d ", i);
if (c++ == 10) {
c = 0;
System.out.println();
}
}
}
System.out.println();
}
}
Output:
337    355    373    535    553    733   2227   2272   2335   2353   2533
2722   3235   3253   3325   3352   3523   3532   5233   5323   5332   7222
22225  22252  22333  22522  23233  23323  23332  25222  32233  32323  32332
33223  33232  33322  52222 222223 222232 222322 223222 232222 322222

## JavaScript

As an unfold, in the recursive pattern described by Nigel Galloway on the Talk page.

(() => {
'use strict';

// ---- NUMBERS WITH PRIME DIGITS WHOSE SUM IS 13 ----

// primeDigitsSummingToN :: Int -> [Int]
const primeDigitsSummingToN = n => {
const primeDigits = [2, 3, 5, 7];
const go = xs =>
fanArrow(
concatMap( // Harvested,
nv => n === nv[1] ? (
[unDigits(nv[0])]
) : []
)
)(
concatMap( // Pruned.
nv => pred(n) > nv[1] ? (
[nv[0]]
) : []
)
)(
// Existing numbers with prime digits appended,
// tupled with the resulting digit sums.
xs.flatMap(
ds => primeDigits.flatMap(d => [
fanArrow(identity)(sum)(
ds.concat(d)
)
])
)
);
return concat(
unfoldr(
xs => 0 < xs.length ? (
Just(go(xs))
) : Nothing()
)(
primeDigits.map(pureList)
)
);
}

// ---------------------- TEST -----------------------
// main :: IO ()
const main = () =>
chunksOf(6)(
primeDigitsSummingToN(13)
).forEach(
x => console.log(x)
)

// ---------------- GENERIC FUNCTIONS ----------------

// Just :: a -> Maybe a
const Just = x => ({
type: 'Maybe',
Nothing: false,
Just: x
});

// Nothing :: Maybe a
const Nothing = () => ({
type: 'Maybe',
Nothing: true,
});

// Tuple (,) :: a -> b -> (a, b)
const Tuple = a =>
b => ({
type: 'Tuple',
'0': a,
'1': b,
length: 2
});

// chunksOf :: Int -> [a] -> [[a]]
const chunksOf = n =>
xs => enumFromThenTo(0)(n)(
xs.length - 1
).reduce(
(a, i) => a.concat([xs.slice(i, (n + i))]),
[]
);

// concat :: [[a]] -> [a]
// concat :: [String] -> String
const concat = xs => (
ys => 0 < ys.length ? (
ys.every(Array.isArray) ? (
[]
) : ''
).concat(...ys) : ys
)(list(xs));

// concatMap :: (a -> [b]) -> [a] -> [b]
const concatMap = f =>
xs => xs.flatMap(f)

// enumFromThenTo :: Int -> Int -> Int -> [Int]
const enumFromThenTo = x1 =>
x2 => y => {
const d = x2 - x1;
return Array.from({
length: Math.floor(y - x2) / d + 2
}, (_, i) => x1 + (d * i));
};

// fanArrow (&&&) :: (a -> b) -> (a -> c) -> (a -> (b, c))
const fanArrow = f =>
// A function from x to a tuple of (f(x), g(x))
// ((,) . f <*> g)
g => x => Tuple(f(x))(
g(x)
);

// identity :: a -> a
const identity = x =>
// The identity function.
x;

// list :: StringOrArrayLike b => b -> [a]
const list = xs =>
// xs itself, if it is an Array,
// or an Array derived from xs.
Array.isArray(xs) ? (
xs
) : Array.from(xs || []);

// pred :: Enum a => a -> a
const pred = x =>
x - 1;

// pureList :: a -> [a]
const pureList = x => [x];

// sum :: [Num] -> Num
const sum = xs =>
// The numeric sum of all values in xs.
xs.reduce((a, x) => a + x, 0);

// unDigits :: [Int] -> Int
const unDigits = ds =>
// The integer with the given digits.
ds.reduce((a, x) => 10 * a + x, 0);

// The 'unfoldr' function is a *dual* to 'foldr':
// while 'foldr' reduces a list to a summary value,
// 'unfoldr' builds a list from a seed value.

// unfoldr :: (b -> Maybe (a, b)) -> b -> [a]
const unfoldr = f =>
v => {
const xs = [];
let xr = [v, v];
while (true) {
const mb = f(xr[1]);
if (mb.Nothing) {
return xs;
} else {
xr = mb.Just;
xs.push(xr[0]);
}
}
};

return main();
})();
Output:
337,355,373,535,553,733
2227,2272,2335,2353,2533,2722
3235,3253,3325,3352,3523,3532
5233,5323,5332,7222,22225,22252
22333,22522,23233,23323,23332,25222
32233,32323,32332,33223,33232,33322
52222,222223,222232,222322,223222,232222
322222

## jq

Works with: jq

Works with gojq, the Go implementation of jq

The first two solutions presented in this section focus on the specific task posed in the title of this page, and the third is a fast analytic solution for determining the count of distinct numbers with prime digits whose sum is any given number. This third solution requires gojq for accuracy if the target sum is relatively large.

All three solutions are based on the observation that the only decimal digits which are prime are [2, 3, 5, 7].

To save space, the first two solutions only present the count of the number of solutions; this is done using the stream counter:

def count(s): reduce s as \$_ (0; .+1);

#### Simple Generate-and-Test Solution

# Output: a stream
def simple:
range(2; 7) as \$n
| [2, 3, 5, 7]
| combinations(\$n)
| join("") | tonumber;

count(simple)
Output:
43

#### A Faster Solution

def faster:
def digits: [2, 3, 5, 7];
def wide(\$max):
def d: digits[] | select(. <= \$max);
if . == 1 then d | [.]
else d as \$first
| (. - 1 | wide(\$max - \$first)) as \$next
| [\$first] + \$next
end;
range(2; 7)
| wide(13)
| join("") | tonumber;

count(faster)

Output:
43

#### Fast Computation of the Count of Numbers

As indicated above, this third solution is analytical (combinatoric), and requires gojq for accuracy for relatively large target sums.

# Input should be a sorted array of distinct positive integers
# Output is a stream of distinct arrays, each of which is sorted, and each sum of which is \$sum
def sorted_combinations(\$sum):
if \$sum <= 0 or length == 0 or \$sum < .[0] then empty
else range(0; length ) as \$i
| .[\$i] as \$x
| ((\$sum / \$x) | floor) as \$maxn
| range(1; 1 + \$maxn) as \$n
| ([range(0; \$n) | \$x]) as \$prefix
| (\$prefix | add // 0 ) as \$psum
| if \$psum == \$sum then \$prefix
else \$prefix + (.[\$i+1 :] | sorted_combinations(\$sum - \$psum) )
end
end;

def factorial: reduce range(2;.+1) as \$i (1; . * \$i);

def product_of_factorials:
reduce .[] as \$n (1; . * (\$n|factorial));

# count the number of distinct permutations
def count_distinct_permutations:
def histogram:
reduce .[] as \$i ([]; .[\$i] += 1);
(length|factorial) / (histogram|product_of_factorials);

def number_of_interesting_numbers(\$total):
def digits: [2, 3, 5, 7];
reduce (digits | sorted_combinations(\$total)) as \$pattern (0;
. + (\$pattern|count_distinct_permutations));

number_of_interesting_numbers(13),
number_of_interesting_numbers(199)
Output:
43
349321957098598244959032342621956

## Julia

using Combinatorics, Primes

function primedigitsums(targetsum)

possibles = mapreduce(x -> fill(x, div(targetsum, x)), vcat, [2, 3, 5, 7])

a = map(x -> evalpoly(BigInt(10), x),
mapreduce(x -> unique(collect(permutations(x))), vcat,
unique(filter(x -> sum(x) == targetsum, collect(combinations(possibles))))))

println("There are \$(length(a)) prime-digit-only numbers summing to \$targetsum : \$a")

end

foreach(primedigitsums, [5, 7, 11, 13])

function primedigitcombos(targetsum)
possibles = [2, 3, 5, 7]
found = Vector{Vector{Int}}()
combos = [Int[]]
tempcombos = Vector{Vector{Int}}()
newcombos = Vector{Vector{Int}}()
while !isempty(combos)
for combo in combos, j in possibles
csum = sum(combo) + j
if csum <= targetsum
newcombo = sort!([combo; j])
csum < targetsum && !(newcombo in newcombos) && push!(newcombos, newcombo)
csum == targetsum && !(newcombo in found) && push!(found, newcombo)
end
end
empty!(combos)
tempcombos = combos
combos = newcombos
newcombos = tempcombos
end
return found
end

function countprimedigitsums(targetsum)
found = primedigitcombos(targetsum)
total = sum(arr -> factorial(BigInt(length(arr))) ÷
prod(x -> factorial(BigInt(count(y -> y == x, arr))), unique(arr)), found)
println("There are \$total prime-digit-only numbers summing to \$targetsum.")
end

foreach(countprimedigitsums, nextprimes(17, 40))

Output:
There are 3 prime-digit-only numbers summing to 5 : [5, 32, 23]
There are 6 prime-digit-only numbers summing to 7 : [7, 52, 25, 322, 232, 223]
There are 19 prime-digit-only numbers summing to 11 : [722, 272, 227, 533, 353, 335, 5222, 2522, 2252, 2225, 3332, 3323, 3233, 2333, 32222, 23222, 22322, 22232, 22223]
There are 43 prime-digit-only numbers summing to 13 : [733, 373, 337, 553, 535, 355, 7222, 2722, 2272, 2227, 5332, 3532, 3352, 5323, 3523, 5233, 2533, 3253, 2353, 3325, 3235, 2335, 52222, 25222, 22522, 22252, 22225, 33322, 33232, 32332, 23332, 33223, 32323, 23323, 32233, 23233, 22333, 322222, 232222, 223222, 222322, 222232, 222223]
There are 221 prime-digit-only numbers summing to 17.
There are 468 prime-digit-only numbers summing to 19.
There are 2098 prime-digit-only numbers summing to 23.
There are 21049 prime-digit-only numbers summing to 29.
There are 45148 prime-digit-only numbers summing to 31.
There are 446635 prime-digit-only numbers summing to 37.
There are 2061697 prime-digit-only numbers summing to 41.
There are 4427752 prime-digit-only numbers summing to 43.
There are 20424241 prime-digit-only numbers summing to 47.
There are 202405001 prime-digit-only numbers summing to 53.
There are 2005642061 prime-digit-only numbers summing to 59.
There are 4307930784 prime-digit-only numbers summing to 61.
There are 42688517778 prime-digit-only numbers summing to 67.
There are 196942068394 prime-digit-only numbers summing to 71.
There are 423011795680 prime-digit-only numbers summing to 73.
There are 4191737820642 prime-digit-only numbers summing to 79.
There are 19338456915087 prime-digit-only numbers summing to 83.
There are 191629965405641 prime-digit-only numbers summing to 89.
There are 4078672831913824 prime-digit-only numbers summing to 97.
There are 18816835854129198 prime-digit-only numbers summing to 101.
There are 40416663565084464 prime-digit-only numbers summing to 103.
There are 186461075642340151 prime-digit-only numbers summing to 107.
There are 400499564627237889 prime-digit-only numbers summing to 109.
There are 1847692833654336940 prime-digit-only numbers summing to 113.
There are 389696778451488128521 prime-digit-only numbers summing to 127.
There are 1797854500757846669066 prime-digit-only numbers summing to 131.
There are 17815422682488317051838 prime-digit-only numbers summing to 137.
There are 38265729200380568226735 prime-digit-only numbers summing to 139.
There are 1749360471151229472803187 prime-digit-only numbers summing to 149.
There are 3757449669085729778349997 prime-digit-only numbers summing to 151.
There are 37233577041224219717325533 prime-digit-only numbers summing to 157.
There are 368957506121989278337474430 prime-digit-only numbers summing to 163.
There are 1702174484494837917764813972 prime-digit-only numbers summing to 167.
There are 16867303726643249517987636148 prime-digit-only numbers summing to 173.
There are 167142638782573042636172836062 prime-digit-only numbers summing to 179.
There are 359005512666242240589945886415 prime-digit-only numbers summing to 181.
There are 16412337250779890525195727788488 prime-digit-only numbers summing to 191.
There are 35252043354611887665339338710961 prime-digit-only numbers summing to 193.
There are 162634253887997896351270835136345 prime-digit-only numbers summing to 197.
There are 349321957098598244959032342621956 prime-digit-only numbers summing to 199.

## Kotlin

Translation of: D
fun primeDigitsSum13(n: Int): Boolean {
var nn = n
var sum = 0
while (nn > 0) {
val r = nn % 10
if (r != 2 && r != 3 && r != 5 && r != 7) {
return false
}
nn /= 10
sum += r
}
return sum == 13
}

fun main() {
// using 2 for all digits, 6 digits is the max prior to over-shooting 13
var c = 0
for (i in 1 until 1000000) {
if (primeDigitsSum13(i)) {
print("%6d ".format(i))
if (c++ == 10) {
c = 0
println()
}
}
}
println()
}
Output:
337    355    373    535    553    733   2227   2272   2335   2353   2533
2722   3235   3253   3325   3352   3523   3532   5233   5323   5332   7222
22225  22252  22333  22522  23233  23323  23332  25222  32233  32323  32332
33223  33232  33322  52222 222223 222232 222322 223222 232222 322222

## Lua

Translation of: C
function prime_digits_sum_13(n)
local sum = 0
while n > 0 do
local r = n % 10
if r ~= 2 and r ~= 3 and r ~= 5 and r ~= 7 then
return false
end
n = math.floor(n / 10)
sum = sum + r
end
return sum == 13
end

local c = 0
for i=1,999999 do
if prime_digits_sum_13(i) then
io.write(string.format("%6d ", i))
if c == 10 then
c = 0
print()
else
c = c + 1
end
end
end
print()
Output:
337    355    373    535    553    733   2227   2272   2335   2353   2533
2722   3235   3253   3325   3352   3523   3532   5233   5323   5332   7222
22225  22252  22333  22522  23233  23323  23332  25222  32233  32323  32332
33223  33232  33322  52222 222223 222232 222322 223222 232222 322222

## Nim

import math, sequtils, strutils

type Digit = 0..9

proc toInt(s: seq[Digit]): int =
## Convert a sequence of digits to an integer.
for n in s:
result = 10 * result + n

const PrimeDigits = @[Digit 2, 3, 5, 7]

var list = PrimeDigits.mapIt(@[it]) # List of sequences of digits.
var result: seq[int]
while list.len != 0:
var nextList: seq[seq[Digit]] # List with one more digit.
for digitSeq in list:
let currSum = sum(digitSeq)
for n in PrimeDigits:
let newSum = currSum + n
let newDigitSeq = digitSeq & n
if newSum < 13: nextList.add newDigitSeq
elif newSum == 13: result.add newDigitSeq.toInt
else: break
list = move(nextList)

for i, n in result:
stdout.write (\$n).align(6), if (i + 1) mod 9 == 0: '\n' else: ' '
echo()
Output:
337    355    373    535    553    733   2227   2272   2335
2353   2533   2722   3235   3253   3325   3352   3523   3532
5233   5323   5332   7222  22225  22252  22333  22522  23233
23323  23332  25222  32233  32323  32332  33223  33232  33322
52222 222223 222232 222322 223222 232222 322222

## Pascal

Works with: Free Pascal
Only counting.
Extreme fast in finding the sum of primesdigits = value.
Limited by Uint64
program PrimSumUpTo13;
{\$IFDEF FPC}
{\$MODE DELPHI}
{\$ELSE}
{\$APPTYPE CONSOLE}
{\$ENDIF}
uses
sysutils;

type
tDigits = array[0..3] of Uint32;
const
MAXNUM = 113;
var
gblPrimDgtCnt :tDigits;
gblCount: NativeUint;

function isPrime(n: NativeUint):boolean;
var
i : NativeUInt;
Begin
result := (n>1);
if n<4 then
EXIT;
result := false;
if n AND 1 = 0 then
EXIT;
i := 3;
while i*i<= n do
Begin
If n MOD i = 0 then
EXIT;
inc(i,2);
end;
result := true;
end;

procedure Sort(var t:tDigits);
var
i,j,k: NativeUint;
temp : Uint32;
Begin
For k := 0 to high(tdigits)-1 do
Begin
temp:= t[k];
j := k;
For i := k+1 to high(tdigits) do
Begin
if temp < t[i] then
Begin
temp := t[i];
j := i;
end;
end;
t[j] := t[k];
t[k] := temp;
end;
end;

function CalcPermCount:NativeUint;
//TempDgtCnt[0] = 3 and TempDgtCnt[1..3]= 2 -> dgtcnt = 3+3*2= 9
//permcount = dgtcnt! /(TempDgtCnt[0]!*TempDgtCnt[1]!*TempDgtCnt[2]!*TempDgtCnt[3]!);
//nom of n! = 1,2,3, 4,5, 6,7, 8,9
//denom = 1,2,3, 1,2, 1,2, 1,2
var
TempDgtCnt : tdigits;
i,f : NativeUint;
begin
TempDgtCnt := gblPrimDgtCnt;
Sort(TempDgtCnt);
//jump over 1/1*2/2*3/3*4/4*..* TempDgtCnt[0]/TempDgtCnt[0]
f := TempDgtCnt[0]+1;
result :=1;

For i := 1 to TempDgtCnt[1] do
Begin
result := (result*f) DIV i;
inc(f);
end;
For i := 1 to TempDgtCnt[2] do
Begin
result := (result*f) DIV i;
inc(f);
end;
For i := 1 to TempDgtCnt[3] do
Begin
result := (result*f) DIV i;
inc(f);
end;
end;

procedure check32(sum3 :NativeUint);
var
n3 : nativeInt;
begin
n3 := sum3 DIV 3;
gblPrimDgtCnt[1]:= 0;
while n3 >= 0 do
begin
//divisible by 2
if sum3 AND 1 = 0 then
Begin
gblPrimDgtCnt[0] := sum3 shr 1;
inc(gblCount,CalcPermCount);
sum3 -= 3;
inc(gblPrimDgtCnt[1]);
dec(n3);
end;
sum3 -= 3;
inc(gblPrimDgtCnt[1]);
dec(n3);
end;
end;

var
Num : NativeUint;
i,sum7,sum5: NativeInt;
BEGIN
writeln('Sum':6,'Count of arrangements':25);

Num := 1;
repeat
inc(num);
if Not(isPrime(Num)) then
CONTINUE;
gblCount := 0;
sum7 :=num;
gblPrimDgtCnt[3] := 0;
while sum7 >=0 do
Begin
sum5 := sum7;
gblPrimDgtCnt[2]:=0;
while sum5 >= 0 do
Begin
check32(sum5);
dec(sum5,5);
inc(gblPrimDgtCnt[2]);
end;
inc(gblPrimDgtCnt[3]);
dec(sum7,7);
end;
writeln(num:6,gblCount:25,' ');
until num > MAXNUM;
END.
Output:
Sum    Count of arrangements
2                        1
3                        1
5                        3
7                        6
11                       19
13                       43
17                      221
19                      468
23                     2098
29                    21049
31                    45148
37                   446635
41                  2061697
43                  4427752
47                 20424241
53                202405001
59               2005642061
61               4307930784
67              42688517778
71             196942068394
73             423011795680
79            4191737820642
83           19338456915087
89          191629965405641
97         4078672831913824
101        18816835854129198
103        40416663565084464
107       186461075642340151
109       400499564627237889
113      1847692833654336940
real  0m0,003s

### using gmp

program PrimSumUpTo13_GMP;
{\$IFDEF FPC}
{\$OPTIMIZATION ON,ALL}
{\$MODE DELPHI}
{\$ELSE}
{\$APPTYPE CONSOLE}
{\$ENDIF}
uses
sysutils,gmp;

type
tDigits = array[0..3] of Uint32;
const
MAXNUM = 199;//999
var
//split factors of n! in Uint64 groups
Fakul : array[0..MAXNUM] of UInt64;
IdxLimits : array[0..MAXNUM DIV 3] of word;
gblPrimDgtCnt :tDigits;

gblSum,
gbldelta : MPInteger; // multi precision (big) integers selve cleaning
s : AnsiString;

procedure Init;
var
i,j,tmp,n :NativeUint;
Begin
//generate n! by Uint64 factors
j := 1;
n := 0;
For i := 1 to MAXNUM do
begin
tmp := j;
j *= i;
if j div i <> tmp then
Begin
IdxLimits[n]:= i-1;
j := i;
inc(n);
end;
Fakul[i] := j;
end;

setlength(s,512);// 997 -> 166
z_init_set_ui(gblSum,0);
z_init_set_ui(gbldelta,0);
end;

function isPrime(n: NativeUint):boolean;
var
i : NativeUInt;
Begin
result := (n>1);
if n<4 then
EXIT;
result := false;
if n AND 1 = 0 then
EXIT;
i := 3;
while i*i<= n do
Begin
If n MOD i = 0 then
EXIT;
inc(i,2);
end;
result := true;
end;

procedure Sort(var t:tDigits);
// sorting descending to reduce calculations
var
i,j,k: NativeUint;
temp : Uint32;
Begin
For k := 0 to high(tdigits)-1 do
Begin
temp:= t[k];
j := k;
For i := k+1 to high(tdigits) do
Begin
if temp < t[i] then
Begin
temp := t[i];
j := i;
end;
end;
t[j] := t[k];
t[k] := temp;
end;
end;

function calcOne(f,n: NativeUint):NativeUint;
var
i,idx,MaxMulLmt : NativeUint;
Begin
result := f;
if n = 0 then
EXIT;

MaxMulLmt := High(MaxMulLmt) DIV (f+n);
z_mul_ui(gbldelta,gbldelta,result);
inc(result);
if n > 1 then
Begin
//multiply by parts of (f+n)!/f! with max Uint64 factors
i := 2;
while (i<=n) do
begin
idx := 1;
while (i<=n) AND (idx<MaxMulLmt) do
Begin
idx *= result;
inc(i);
inc(result);
end;
z_mul_ui(gbldelta,gbldelta,idx);
end;

//divide by n! with max Uint64 divisors
idx := 0;
if n > IdxLimits[idx] then
repeat
z_divexact_ui(gbldelta,gbldelta,Fakul[IdxLimits[idx]]);
inc(idx);
until IdxLimits[idx] >= n;
z_divexact_ui(gbldelta,gbldelta,Fakul[n]);
end;
end;

procedure CalcPermCount;
//TempDgtCnt[0] = 3 and TempDgtCnt[1..3]= 2 -> dgtcnt = 3+3*2= 9
//permcount = dgtcnt! /(TempDgtCnt[0]!*TempDgtCnt[1]!*TempDgtCnt[2]!*TempDgtCnt[3]!);
//nom of n! = 1,2,3, 4,5, 6,7, 8,9
//denom = 1,2,3, 1,2, 1,2, 1,2
var
TempDgtCnt : tdigits;
f : NativeUint;
begin
TempDgtCnt := gblPrimDgtCnt;
Sort(TempDgtCnt);
//jump over 1/1*2/2*3/3*4/4*..*
//res := 1;
f := TempDgtCnt[0]+1;
z_set_ui(gbldelta,1);

f := calcOne(f,TempDgtCnt[1]);
f := calcOne(f,TempDgtCnt[2]);
f := calcOne(f,TempDgtCnt[3]);

end;

procedure check32(sum3 :NativeInt);
var
n3 : nativeInt;
begin
n3 := sum3 DIV 3;
gblPrimDgtCnt[1]:= 0;
while n3 >= 0 do
begin
//divisible by 2
if sum3 AND 1 = 0 then
Begin
gblPrimDgtCnt[0] := sum3 shr 1;
CalcPermCount;
end;
sum3 -= 3;
inc(gblPrimDgtCnt[1]);
dec(n3);
end;
end;
procedure CheckAll(num:NativeUint);
var
sum7,sum5: NativeInt;
BEGIN
z_set_ui(gblSum,0);
sum7 :=num;
gblPrimDgtCnt[3] := 0;
while sum7 >=0 do
Begin
sum5 := sum7;
gblPrimDgtCnt[2]:=0;
while sum5 >= 0 do
Begin
check32(sum5);
dec(sum5,5);
inc(gblPrimDgtCnt[2]);
end;
inc(gblPrimDgtCnt[3]);
dec(sum7,7);
end;
end;

var
T0 : Int64;
Num : NativeUint;
BEGIN
Init;

T0 := GettickCount64;
// Number crunching goes here
writeln('Sum':6,'Count of arrangements':25);
For num := 2 to MAXNUM do
IF isPrime(Num) then
Begin
CheckAll(num);

z_get_str(pChar(s),10,gblSum);
writeln(num:6,' ',pChar(s));
end;
writeln('Time taken : ',GettickCount64-T0,' ms');

z_clear(gblSum);
z_clear(gbldelta);
END.
Output:
tio.run/#pascal-fpc
Sum    Count of arrangements
2  1
3  1
5  3
7  6
11  19
13  43
17  221
19  468
23  2098
29  21049
31  45148
37  446635
41  2061697
43  4427752
47  20424241
53  202405001
59  2005642061
61  4307930784
67  42688517778
71  196942068394
73  423011795680
79  4191737820642
83  19338456915087
89  191629965405641
97  4078672831913824
101  18816835854129198
103  40416663565084464
107  186461075642340151
109  400499564627237889
113  1847692833654336940
127  389696778451488128521
131  1797854500757846669066
137  17815422682488317051838
139  38265729200380568226735
149  1749360471151229472803187
151  3757449669085729778349997
157  37233577041224219717325533
163  368957506121989278337474430
167  1702174484494837917764813972
173  16867303726643249517987636148
179  167142638782573042636172836062
181  359005512666242240589945886415
191  16412337250779890525195727788488
193  35252043354611887665339338710961
197  162634253887997896351270835136345
199  349321957098598244959032342621956
Time taken : 23 ms

## Perl

#!/usr/bin/perl

use strict;
use warnings;

my @queue = my @primedigits = ( 2, 3, 5, 7 );
my \$numbers;

while( my \$n = shift @queue )
{
if( eval \$n == 13 )
{
\$numbers .= \$n =~ tr/+//dr . " ";
}
elsif( eval \$n < 13 )
{
push @queue, map "\$n+\$_", @primedigits;
}
}
print \$numbers =~ s/.{1,80}\K /\n/gr;
Output:
337 355 373 535 553 733 2227 2272 2335 2353 2533 2722 3235 3253 3325 3352 3523
3532 5233 5323 5332 7222 22225 22252 22333 22522 23233 23323 23332 25222 32233
32323 32332 33223 33232 33322 52222 222223 222232 222322 223222 232222 322222

## Phix

with javascript_semantics
function unlucky(sequence set, integer needed, string v="", sequence res={})
if needed=0 then
res = append(res,sprintf("%6s",v))
elsif needed>0 then
for i=length(set) to 1 by -1 do
res = unlucky(set,needed-set[i],(set[i]+'0')&v,res)
end for
end if
return res
end function

sequence r = sort(unlucky({2,3,5,7},13))
puts(1,join_by(r,1,11," "))
Output:
337    355    373    535    553    733   2227   2272   2335   2353   2533
2722   3235   3253   3325   3352   3523   3532   5233   5323   5332   7222
22225  22252  22333  22522  23233  23323  23332  25222  32233  32323  32332
33223  33232  33322  52222 222223 222232 222322 223222 232222 322222

### iterative

Queue-based version of Nigel's recursive algorithm, same output.

with javascript_semantics
requires("0.8.2") -- uses latest apply() mods, rest is fine
constant dgts = {2,3,5,7}
function unlucky()
sequence res = {}, q = {{0,0}}
integer s, -- partial digit sum, <=11
v  -- corresponding value
while length(q) do
{{s,v}, q} = {q[1], q[2..\$]}
for i=1 to length(dgts) do
integer d = dgts[i], {ns,nv} = {s+d,v*10+d}
if ns<=11 then q &= {{ns,nv}}
elsif ns=13 then res &= nv end if
end for
end while
return res
end function

sequence r = unlucky()
r = apply(true,sprintf,{{"%6d"},r})
puts(1,join_by(r,1,11," "))

I've archived a slightly more OTT version: Numbers_with_prime_digits_whose_sum_is_13/Phix.

## Prolog

digit_sum(N, M) :- digit_sum(N, 0, M).
digit_sum(0, A, B) :- !, A = B.
digit_sum(N, A0, M) :-
divmod(N, 10, Q, R),
plus(A0, R, A1),
digit_sum(Q, A1, M).

prime_digits(0).
prime_digits(N) :-
prime_digits(M),
member(D, [2, 3, 5, 7]),
N is 10 * M + D.

prime13(N) :-
prime_digits(N),
(N > 333_333 -> !, false ; true),
digit_sum(N, 13).

main :-
findall(N, prime13(N), S),
format("Those numbers whose digits are all prime and sum to 13 are: ~n~w~n", [S]),
halt.

?- main.

Output:
Those numbers whose digits are all prime and sum to 13 are:
[337,355,373,535,553,733,2227,2272,2335,2353,2533,2722,3235,3253,3325,3352,3523,3532,5233,5323,5332,7222,22225,22252,22333,22522,23233,23323,23332,25222,32233,32323,32332,33223,33232,33322,52222,222223,222232,222322,223222,232222,322222]

## Raku

put join ', ', sort +*, unique flat
< 2 2 2 2 2 3 3 3 5 5 7 >.combinations
.grep( *.sum == 13 )
.map( { .join => \$_ } )
.map: { .value.permutations».join }
Output:
337, 355, 373, 535, 553, 733, 2227, 2272, 2335, 2353, 2533, 2722, 3235, 3253, 3325, 3352, 3523, 3532, 5233, 5323, 5332, 7222, 22225, 22252, 22333, 22522, 23233, 23323, 23332, 25222, 32233, 32323, 32332, 33223, 33232, 33322, 52222, 222223, 222232, 222322, 223222, 232222, 322222

## REXX

/*REXX pgm finds and displays all decimal numbers whose digits are prime and sum to 13. */
parse arg LO HI COLS . /*obtain optional arguments from the CL*/
if LO=='' | LO=="," then LO= 337 /*Not specified? Then use the default.*/
if HI=='' | HI=="," then HI= 322222 /* " " " " " " */
if cols=='' | cols=="," then cols= 10 /* " " " " " " */
w= 10 /*width of a number in any column. */
title= ' decimal numbers found whose digits are prime and sum to 13 '
say ' index │'center(title, 1 + cols*(w+1) )
say '───────┼'center("" , 1 + cols*(w+1), '─')
w= 10 /*max width of a integer in any column.*/
found= 0; idx= 1 /*the number of numbers found (so far).*/
\$= /*variable to hold the list of #s found*/
do j=LO for HI-LO+1 /*search for numbers in this range. */
if verify(j, 2357) \== 0 then iterate /*J must be comprised of prime digits.*/
parse var j a 2 b 3 '' -1 z /*parse: 1st, 2nd, & last decimal digs.*/
sum= a + b + z /*sum: " " " " " " */
do k=3 for length(j)-3 /*only need to sum #s with #digits ≥ 4 */
sum= sum + substr(j, k, 1) /*sum some middle decimal digits of J.*/
end /*k*/
if sum\==13 then iterate /*Sum not equal to 13? Then skip this #*/
found= found + 1 /*bump the number of numbers found. */
c= commas(j) /*maybe add commas to the number. */
\$= \$ right( commas(j), w) /*add the found number ───► the \$ list.*/
if found//cols\==0 then iterate /*have we populated a line of output? */
say center(idx, 7)'│' substr(\$, 2); \$= /*display what we have so far (cols). */
idx= idx + cols /*bump the index count for the output*/
end /*j*/

if \$\=='' then say center(idx, 7)"│" substr(\$, 2) /*possible display residual output.*/
say '───────┴'center("" , 1 + cols*(w+1), '─')
say
say 'Found ' commas(found) title
exit 0 /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
commas: parse arg _; do jc=length(_)-3 to 1 by -3; _=insert(',', _, jc); end; return _
output   when using the internal default inputs:
index │                          decimal numbers found whose digits are prime and sum to 13
───────┼───────────────────────────────────────────────────────────────────────────────────────────────────────────────
1   │        337        355        373        535        553        733      2,227      2,272      2,335      2,353
11   │      2,533      2,722      3,235      3,253      3,325      3,352      3,523      3,532      5,233      5,323
21   │      5,332      7,222     22,225     22,252     22,333     22,522     23,233     23,323     23,332     25,222
31   │     32,233     32,323     32,332     33,223     33,232     33,322     52,222    222,223    222,232    222,322
41   │    223,222    232,222    322,222
───────┴───────────────────────────────────────────────────────────────────────────────────────────────────────────────

Found  43  decimal numbers found whose digits are prime and sum to 13

## Ring

sum = 0
limit = 1000000
aPrimes = []

for n = 1 to limit
sum = 0
st = string(n)
for m = 1 to len(st)
num = number(st[m])
if isprime(num)
sum = sum + num
flag = 1
else
flag = 0
exit
ok
next
if flag = 1 and sum = 13
ok
next

see "Unlucky numbers are:" + nl
see showArray(aPrimes)

func showarray vect
svect = ""
for n in vect
svect += "" + n + ","
next
? "[" + left(svect, len(svect) - 1) + "]"

Output:
Unlucky numbers are:
[337,355,373,535,553,733,2227,2272,2335,2353,2533,2722,3235,3253,3325,3352,3523,3532,5233,5323,5332,7222,22225,22252,22333,22522,23233,23323,23332,25222,32233,32323,32332,33223,33232,33322,52222,222223,222232,222322,223222,232222,322222]

## Ruby

Translation of: C
def primeDigitsSum13(n)
sum = 0
while n > 0
r = n % 10
if r != 2 and r != 3 and r != 5 and r != 7 then
return false
end
n = (n / 10).floor
sum = sum + r
end
return sum == 13
end

c = 0
for i in 1 .. 1000000
if primeDigitsSum13(i) then
print "%6d " % [i]
if c == 10 then
c = 0
print "\n"
else
c = c + 1
end
end
end
print "\n"

Output:
337    355    373    535    553    733   2227   2272   2335   2353   2533
2722   3235   3253   3325   3352   3523   3532   5233   5323   5332   7222
22225  22252  22333  22522  23233  23323  23332  25222  32233  32323  32332
33223  33232  33322  52222 222223 222232 222322 223222 232222 322222

## Sidef

func generate_from_prefix(sum, p, base, digits) {

var seq = [p]

for d in (digits) {
seq << __FUNC__(sum, [d, p...], base, digits)... if (p.sum+d <= sum)
}

return seq
}

func numbers_with_digitsum(sum, base = 10, digits = (base-1 -> primes)) {

digits.map {|p| generate_from_prefix(sum, [p], base, digits)... }\
.map {|t| digits2num(t, base) }\
.grep {|t| t.sumdigits(base) == sum }\
.sort
}

say numbers_with_digitsum(13)
Output:
[337, 355, 373, 535, 553, 733, 2227, 2272, 2335, 2353, 2533, 2722, 3235, 3253, 3325, 3352, 3523, 3532, 5233, 5323, 5332, 7222, 22225, 22252, 22333, 22522, 23233, 23323, 23332, 25222, 32233, 32323, 32332, 33223, 33232, 33322, 52222, 222223, 222232, 222322, 223222, 232222, 322222]

## Visual Basic .NET

Translation of: Phix

Same recursive method.

Imports System
Imports System.Console
Imports LI = System.Collections.Generic.SortedSet(Of Integer)

Module Module1
Function unl(ByVal res As LI, ByVal lst As LI, ByVal lft As Integer, ByVal Optional mul As Integer = 1, ByVal Optional vlu As Integer = 0) As LI
If lft = 0 Then
ElseIf lft > 0 Then
For Each itm As Integer In lst
res = unl(res, lst, lft - itm, mul * 10, vlu + itm * mul)
Next
End If
Return res
End Function

Sub Main(ByVal args As String())
WriteLine(string.Join(" ",
unl(new LI From {}, new LI From { 2, 3, 5, 7 }, 13)))
End Sub
End Module
Output:
337 355 373 535 553 733 2227 2272 2335 2353 2533 2722 3235 3253 3325 3352 3523 3532 5233 5323 5332 7222 22225 22252 22333 22522 23233 23323 23332 25222 32233 32323 32332 33223 33232 33322 52222 222223 222232 222322 223222 232222 322222

### Alternate

Thanks to Nigel Galloway's suggestion from the discussion page.

Imports Tu = System.Tuple(Of Integer, Integer)

Module Module1

Sub Main()
Dim w As New List(Of Tu), sum, x As Integer,
lst() As Integer = { 2, 3, 5, 7 }
For Each x In lst : w.Add(New Tu(x, x)) : Next
While w.Count > 0 : With w(0) : For Each j As Integer In lst
sum = .Item2 + j
If sum = 13 Then Console.Write("{0}{1} ", .Item1, j)
If sum < 12 Then w.Add(New Tu(.Item1 * 10 + j, sum))
Next : End With : w.RemoveAt(0) : End While
End Sub

End Module

Same output.

## Wren

Library: Wren-math
Library: Wren-seq
Library: Wren-sort

As the only digits which are prime are [2, 3, 5, 7], it is clear that a number must have between 3 and 6 digits for them to sum to 13.

import "/math" for Nums
import "/seq" for Lst
import "/sort" for Sort

var combrep // recursive
combrep = Fn.new { |n, lst|
if (n == 0 ) return [[]]
if (lst.count == 0) return []
var r = combrep.call(n, lst[1..-1])
for (x in combrep.call(n-1, lst)) {
var y = x.toList
}
return r
}

var permute // recursive
permute = Fn.new { |input|
if (input.count == 1) return [input]
var perms = []
var toInsert = input[0]
for (perm in permute.call(input[1..-1])) {
for (i in 0..perm.count) {
var newPerm = perm.toList
newPerm.insert(i, toInsert)
}
}
return perms
}

var primes = [2, 3, 5, 7]
var res = []
for (n in 3..6) {
var reps = combrep.call(n, primes)
for (rep in reps) {
if (Nums.sum(rep) == 13) {
var perms = permute.call(rep)
for (i in 0...perms.count) perms[i] = Num.fromString(perms[i].join())
}
}
}
Sort.quick(res)
System.print("Those numbers whose digits are all prime and sum to 13 are:")
System.print(res)
Output:
Those numbers whose digits are all prime and sum to 13 are:
[337, 355, 373, 535, 553, 733, 2227, 2272, 2335, 2353, 2533, 2722, 3235, 3253, 3325, 3352, 3523, 3532, 5233, 5323, 5332, 7222, 22225, 22252, 22333, 22522, 23233, 23323, 23332, 25222, 32233, 32323, 32332, 33223, 33232, 33322, 52222, 222223, 222232, 222322, 223222, 232222, 322222]

## XPL0

int N, M, S, D;
[for N:= 2 to 322222 do
[M:= N; S:= 0;
repeat M:= M/10; \get digit D
D:= remainder(0);
case D of
2, 3, 5, 7:
[S:= S+D;
if S=13 and M=0 \all digits included\ then
[IntOut(0, N); ChOut(0, ^ )];
]
other M:= 0; \digit not prime so exit repeat loop
until M=0; \all digits in N tested or digit not prime
];
]
Output:
337 355 373 535 553 733 2227 2272 2335 2353 2533 2722 3235 3253 3325 3352 3523 3532 5233 5323 5332 7222 22225 22252 22333 22522 23233 23323 23332 25222 32233 32323 32332 33223 33232 33322 52222 222223 222232 222322 223222 232222 322222