Sum of square and cube digits of an integer are primes

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Sum of square and cube digits of an integer are primes 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 and show here all positive integers n less than 100 where:

• the sum of the digits of the square of n is prime; and
• the sum of the digits of the cube of n is also prime.

Example

16 satisfies the task descrption because 16 x 16 = 256 has a digit sum of 13 which is prime and 16 x 16 x 16 = 4096 has a digit sum of 19 which is also prime.

ALGOL 68

```BEGIN # find numbers where the digit sums of the square and cube are prtime #
INT max number = 99; # maximum number to consider #
[]BOOL prime = PRIMESIEVE ( INT d sum := 9; # calculate the largest possible digit sum #
INT n     := max number * max number * max number;
WHILE ( n OVERAB 10 ) > 0 DO
d sum +:= 9
OD;
d sum
);
# returns the sum of the digits of n #
OP  DIGITSUM = ( INT n )INT:
BEGIN
INT v      := ABS n;
INT result := v MOD 10;
WHILE ( v OVERAB 10 ) > 0 DO
result +:= v MOD 10
OD;
result
END # DIGITSUM # ;
FOR i TO max number DO
INT i2 = i * i;
IF prime[ DIGITSUM i2 ] THEN
IF prime[ DIGITSUM ( i * i2 ) ] THEN
print( ( " ", whole( i, 0 ) ) )
FI
FI
OD
END```
Output:
``` 16 17 25 28 34 37 47 52 64
```

APL

```(⊢(/⍨)∧/∘((2=0+.=⍳|⊢)∘(+/⍎¨∘⍕)¨*∘2 3)¨)⍳100
```
Output:
`16 17 25 28 34 37 47 52 64`

Arturo

```print select 1..100 'x ->
and? [prime? sum digits x^2]
[prime? sum digits x^3]
```
Output:
`16 17 25 28 34 37 47 52 64`

AWK

```# syntax: GAWK -f SUM_OF_SQUARE_AND_CUBE_DIGITS_OF_AN_INTEGER_ARE_PRIMES.AWK
# converted from FreeBASIC
BEGIN {
start = 1
stop = 99
for (i=start; i<=stop; i++) {
if (is_prime(digit_sum(i^3,10)) && is_prime(digit_sum(i^2,10))) {
printf("%5d%1s",i,++count%10?"":"\n")
}
}
printf("\nSum of square and cube digits are prime %d-%d: %d\n",start,stop,count)
exit(0)
}
function digit_sum(n,b,  s) { # digital sum of n in base b
while (n) {
s += n % b
n = int(n/b)
}
return(s)
}
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:
```   16    17    25    28    34    37    47    52    64
Sum of square and cube digits are prime 1-99: 9
```

BQN

```# 'Library' functions from BQNCrate
Digits ← 10 {⌽𝕗|⌊∘÷⟜𝕗⍟(↕1+·⌊𝕗⋆⁼1⌈⊢)}
Prime ← 2=·+´0=(1+↕)⊸|

(∧˝∘⍉∘((Prime +´∘Digits)¨⋆⌜⟜2‿3))⊸/↕100```
Output:
`⟨ 16 17 25 28 34 37 47 52 64 ⟩`

C

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

int digit_sum(int n) {
int sum;
for (sum = 0; n; n /= 10) sum += n % 10;
return sum;
}

/* The numbers involved are tiny */
bool prime(int n) {
if (n<4) return n>=2;
for (int d=2; d*d <= n; d++)
if (n%d == 0) return false;
return true;
}

int main() {
for (int i=1; i<100; i++)
if (prime(digit_sum(i*i)) & prime(digit_sum(i*i*i)))
printf("%d ", i);
printf("\n");
return 0;
}
```
Output:
`16 17 25 28 34 37 47 52 64`

CLU

```digit_sum = proc (n: int) returns (int)
sum: int := 0
while n>0 do
sum := sum + n // 10
n := n / 10
end
return(sum)
end digit_sum

% The numbers tested for primality are very small,
% so this simple test suffices.
prime = proc (n: int) returns (bool)
if n<2 then return(false) end
d: int := 2
while d*d <= n do
if n//d=0 then return(false) end
d := d+1
end
return(true)
end prime

accept = proc (n: int) returns (bool)
return(prime(digit_sum(n**2)) cand prime(digit_sum(n**3)))
end accept

start_up = proc ()
po: stream := stream\$primary_output()
for i: int in int\$from_to(1, 99) do
if accept(i) then
stream\$puts(po, int\$unparse(i) || " ")
end
end
end start_up```
Output:
`16 17 25 28 34 37 47 52 64`

COBOL

```       IDENTIFICATION DIVISION.
PROGRAM-ID. SQUARE-CUBE-DIGITS-PRIME.

DATA DIVISION.
WORKING-STORAGE SECTION.
01 NUMBER-SEARCH-VARS.
03 CAND             PIC 9(6).
03 SQUARE           PIC 9(6).
03 CUBE             PIC 9(6).

01 SUM-DIGITS-VARS.
03 SUM-NUM          PIC 9(6).
03 DIGITS           PIC 9 OCCURS 6 TIMES INDEXED BY D
REDEFINES SUM-NUM.
03 SUM              PIC 99.

01 PRIME-TEST-VARS.
03 DIVISOR          PIC 99.
03 DIV-TEST         PIC 99V9999.
03 FILLER           REDEFINES DIV-TEST.
05 FILLER        PIC 99.
05 FILLER        PIC 9999.
88 DIVISIBLE  VALUE ZERO.
03 PRIME-FLAG       PIC X.
88 PRIME         VALUE '*'.

01 OUT-FMT             PIC Z9.

PROCEDURE DIVISION.
BEGIN.
PERFORM CHECK-NUMBER VARYING CAND FROM 1 BY 1
UNTIL CAND IS EQUAL TO 100.
STOP RUN.

CHECK-NUMBER.
MULTIPLY CAND BY CAND GIVING SQUARE.
MULTIPLY CAND BY SQUARE GIVING CUBE.
MOVE SQUARE TO SUM-NUM.
PERFORM SUM-DIGITS.
PERFORM PRIME-TEST.
IF PRIME,
MOVE CUBE TO SUM-NUM,
PERFORM SUM-DIGITS,
PERFORM PRIME-TEST,
IF PRIME,
MOVE CAND TO OUT-FMT,
DISPLAY OUT-FMT.

SUM-DIGITS.
MOVE ZERO TO SUM.
PERFORM SUM-DIGIT VARYING D FROM 1 BY 1
UNTIL D IS GREATER THAN 6.

SUM-DIGIT.

PRIME-TEST.
MOVE '*' TO PRIME-FLAG.
PERFORM CHECK-DIVISOR VARYING DIVISOR FROM 2 BY 1
UNTIL NOT PRIME, OR DIVISOR IS EQUAL TO SUM.

CHECK-DIVISOR.
DIVIDE SUM BY DIVISOR GIVING DIV-TEST.
IF DIVISIBLE, MOVE SPACE TO PRIME-FLAG.
```
Output:
```16
17
25
28
34
37
47
52
64```

F#

This task uses Extensible Prime Generator (F#)

```// Sum of square and cube digits of an integer are primes. Nigel Galloway: December 22nd., 2021
let rec fN g=function 0->g |n->fN(g+n%10)(n/10)
[1..99]|>List.filter(fun g->isPrime(fN 0 (g*g)) && isPrime(fN 0 (g*g*g)))|>List.iter(printf "%d "); printfn ""
```
Output:
```16 17 25 28 34 37 47 52 64
```

Factor

Works with: Factor version 0.99 2021-06-02
```USING: kernel math math.functions math.primes math.text.utils prettyprint sequences ;

100 <iota> [ [ sq ] [ 3 ^ ] bi [ 1 digit-groups sum prime? ] both? ] filter .
```
Output:
```V{ 16 17 25 28 34 37 47 52 64 }
```

FOCAL

```01.10 F I=1,100;D 2
01.20 Q

02.10 F P=2,3;S N=I^P;D 3;D 4;I (C)2.3
02.20 T %2,I,!
02.30 R

03.10 S S=0
03.20 S M=FITR(N/10)
03.30 S S=S+(N-M*10)
03.40 S N=M
03.50 I (-N)3.2

04.10 S C=0
04.20 I (1-S)4.3;S C=-1;R
04.30 I (2-S)4.4;S C=0;R
04.40 F D=2,FSQT(S)+1;D 5;I (C)4.5
04.50 R

05.10 S Z=S/D
05.20 I (FITR(Z)-Z)5.3;S C=-1
05.30 R```
Output:
```= 16
= 17
= 25
= 28
= 34
= 37
= 47
= 52
= 64```

FreeBASIC

```function digsum(byval n as uinteger, b as const uinteger) as uinteger
'digital sum of n in base b
dim as integer s
while n
s+=n mod b
n\=b
wend
return s
end function

function isprime(n as const uinteger) as boolean
if n<2 then return false
if n<4 then return true
if n mod 2 = 0 then return false
dim as uinteger i = 3
while i*i<=n
if n mod i = 0 then return false
i+=2
wend
return true
end function

for n as uinteger = 1 to 99
if isprime(digsum(n^3,10)) andalso isprime(digsum(n^2,10)) then print n;"   ";
next n
```
Output:
`16   17   25   28   34   37   47   52   64`

Go

Library: Go-rcu
```package main

import (
"fmt"
"rcu"
)

func main() {
for i := 1; i < 100; i++ {
if !rcu.IsPrime(rcu.DigitSum(i*i, 10)) {
continue
}
if rcu.IsPrime(rcu.DigitSum(i*i*i, 10)) {
fmt.Printf("%d ", i)
}
}
fmt.Println()
}
```
Output:
```16 17 25 28 34 37 47 52 64
```

```import Data.Bifunctor (first)
import Data.Numbers.Primes (isPrime)

---- SQUARE AND CUBE BOTH HAVE PRIME DECIMAL DIGIT SUMS --

p :: Int -> Bool
p =
((&&) . primeDigitSum . (^ 2))
<*> (primeDigitSum . (^ 3))

--------------------------- TEST -------------------------
main :: IO ()
main = print \$ filter p [2 .. 99]

------------------------- GENERIC ------------------------
primeDigitSum :: Int -> Bool
primeDigitSum = isPrime . digitSum 10

digitSum :: Int -> Int -> Int
digitSum base = go
where
go 0 = 0
go n = uncurry (+) . first go \$ quotRem n base
```
Output:
`[16,17,25,28,34,37,47,52,64]`

J

```((1*./@p:[:+/@|:10#.^:_1^&2 3)"0#]) i.100
```
Output:
`16 17 25 28 34 37 47 52 64`

Julia

```using Primes

is_sqcubsumprime(n) = isprime(sum(digits(n*n))) && isprime(sum(digits(n*n*n)))
println(filter(is_sqcubsumprime, 1:100)) # [16, 17, 25, 28, 34, 37, 47, 52, 64]
```

```            NORMAL MODE IS INTEGER

BOOLEAN PRIME
DIMENSION PRIME(100)
PRIME(0)=0B
PRIME(1)=0B
THROUGH SET, FOR P=2, 1, P.G.100
SET         PRIME(P)=1B
THROUGH SIEVE, FOR P=2, 1, P*P.G.100
THROUGH SIEVE, FOR C=P*P, P, C.G.100
SIEVE       PRIME(C)=0B

THROUGH CHECK, FOR I=1, 1, I.GE.100
WHENEVER .NOT.PRIME(DIGSUM.(I*I)), TRANSFER TO CHECK
WHENEVER .NOT.PRIME(DIGSUM.(I*I*I)), TRANSFER TO CHECK
PRINT FORMAT FMT, I
CHECK       CONTINUE

INTERNAL FUNCTION(N)
ENTRY TO DIGSUM.
SUM=0
NN=N
LOOP        WHENEVER NN.G.0
NXT=NN/10
SUM=SUM+NN-NXT*10
NN=NXT
TRANSFER TO LOOP
END OF CONDITIONAL
FUNCTION RETURN SUM
END OF FUNCTION

VECTOR VALUES FMT = \$I2*\$
END OF PROGRAM```
Output:
```16
17
25
28
34
37
47
52
64```

Perl

Library: ntheory
```#!/usr/bin/perl

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

my @results = grep
is_prime( vecsum( split //, \$_ ** 2 ) ) &&
is_prime( vecsum( split //, \$_ ** 3 ) ), 1 .. 100;
print "@results\n";
```
Output:
```16 17 25 28 34 37 47 52 64
```

Phix

```with javascript_semantics
function ipsd(integer n) return is_prime(sum(sq_sub(sprintf("%d",n),'0'))) end function
function scdp(integer n) return ipsd(n*n) and ipsd(n*n*n) end function
pp(filter(tagset(99),scdp))
```
Output:
```{16,17,25,28,34,37,47,52,64}
```

Python

Procedural

```#!/usr/bin/python

def isPrime(n):
for i in range(2, int(n**0.5) + 1):
if n % i == 0:
return False
return True

def digSum(n, b):
s = 0
while n:
s += (n % b)
n = n // b
return s

if __name__ == '__main__':
for n in range(11, 99):
if isPrime(digSum(n**3, 10)) and isPrime(digSum(n**2, 10)):
print(n, end = "  ")
```
Output:
`16  17  25  28  34  37  47  52  64`

Functional

```'''Square and cube both have prime decimal digit sums'''

# p :: Int -> Bool
def p(n):
'''True if the square and the cube of N both have
decimal digit sums which are prime.
'''
return primeDigitSum(n ** 2) and primeDigitSum(n ** 3)

# ------------------------- TEST -------------------------
# main :: IO ()
def main():
'''Matches in the range [1..99]'''
print([
x for x in range(2, 100)
if p(x)
])

# ----------------------- GENERIC ------------------------

# primeDigitSum :: Int -> Bool
def primeDigitSum(n):
'''True if the sum of the decimal digits of n is prime.
'''
return isPrime(digitSum(10)(n))

# digitSum :: Int -> Int
def digitSum(base):
'''The sum of the digits of n in a given base.
'''
def go(n):
q, r = divmod(n, base)
return go(q) + r if n else 0
return go

# isPrime :: Int -> Bool
def isPrime(n):
'''True if n is prime.'''
if n in (2, 3):
return True
if 2 > n or 0 == n % 2:
return False
if 9 > n:
return True
if 0 == n % 3:
return False

def q(x):
return 0 == n % x or 0 == n % (2 + x)

return not any(map(q, range(5, 1 + int(n ** 0.5), 6)))

# MAIN ---
if __name__ == '__main__':
main()
```
Output:
`[16, 17, 25, 28, 34, 37, 47, 52, 64]`

Quackery

`isprime` is defined at Primality by trial division#Quackery.

```  [ 0 swap
[ dup while
10 /mod
rot + swap
again ]
drop ]        is digitsum ( n --> n )

98 times
[ i^ 1+ 2 **
digitsum isprime if
[  i^ 1+ 3 **
digitsum isprime if
[ i^ 1+ echo sp ] ] ]```
Output:
`16 17 25 28 34 37 47 52 64`

Raku

```say ^100 .grep: { .².comb.sum.is-prime && .³.comb.sum.is-prime }
```
Output:
`(16 17 25 28 34 37 47 52 64)`

Ring

```load "stdlib.ring"
see "working..." +nl

limit = 100

for n = 1 to limit
sums = 0
sumc = 0
sps = string(pow(n,2))
spc = string(pow(n,3))
for m = 1 to len(sps)
sums = sums + sps[m]
next
for p = 1 to len(spc)
sumc = sumc + spc[p]
next
if isprime(sums) and isprime(sumc)
see "" + n + " "
ok
next

see nl + "done..." + nl```
Output:
```working...
16 17 25 28 34 37 47 52 64
done...
```

Sidef

```1..99 -> grep { .square.digits_sum.is_prime && .cube.digits_sum.is_prime }.say
```
Output:
```[16, 17, 25, 28, 34, 37, 47, 52, 64]
```

TinyBASIC

This can only go up to 31 because 32^3 is too big to fit in a signed 16-bit int.

```REM      N, the number to be tested
REM      D, the digital sum of its square or cube
REM      T, temporary variable
REM      Z, did D test as prime or not

LET N = 1
10 LET T = N*N*N
GOSUB 20
GOSUB 30
IF Z = 0 THEN GOTO 11
LET T = N*N
GOSUB 20
GOSUB 30
IF Z = 0 THEN GOTO 11
PRINT N
11 IF N = 31 THEN END
LET N = N + 1
GOTO 10
20 LET D = 0
21 IF T = 0 THEN RETURN
LET D = D + (T-(T/10)*10)
LET T = T/10
GOTO 21
30 LET Z = 0
IF D < 2 THEN RETURN
LET Z = 1
IF D < 4 THEN RETURN
LET Z = 0
IF (D/2)*2 = D THEN RETURN
LET T = 1
31 LET T = T + 2
IF T*T>D THEN GOTO 32
IF (D/T)*T=D THEN RETURN
GOTO 31
32 LET Z = 1
RETURN```
Output:
```
16
17
25

28```

Wren

Library: Wren-math
```import "./math" for Int

for (i in 1..99) {
if (Int.isPrime(Int.digitSum(i*i)) && Int.isPrime(Int.digitSum(i*i*i))) System.write("%(i) ")
}
System.print()
```
Output:
```16 17 25 28 34 37 47 52 64
```

XPL0

```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 SumDigits(N);      \Return the sum of digits in N
int  N, Sum;
[Sum:= 0;
while N do
[N:= N/10;
Sum:= Sum + rem(0);
];
return Sum;
];

int N;
[for N:= 0 to 100-1 do
if IsPrime(SumDigits(N*N)) & IsPrime(SumDigits(N*N*N)) then
[IntOut(0, N);  ChOut(0, ^ )];
]```
Output:
```16 17 25 28 34 37 47 52 64
```

Yabasic

Translation of: Ring
```// Rosetta Code problem: http://rosettacode.org/wiki/Sum_of_square_and_cube_digits_of_an_integer_are_primes
// by Galileo, 04/2022

sub isPrime(n)
local i

if n < 4 return n >= 2
for i = 2 to sqrt(n)
if not mod(n, i) return false
next
return true
end sub

limit = 100

for n = 1 to limit
sums = 0
sumc = 0
sps\$ = str\$(n^2)
spc\$ = str\$(n^3)
for m = 1 to len(sps\$)
sums = sums + val(mid\$(sps\$, m, 1))
next
for p = 1 to len(spc\$)
sumc = sumc + val(mid\$(spc\$, p, 1))
next
if isPrime(sums) and isPrime(sumc) then
print n, " ";
endif
next
print```
Output:
```16 17 25 28 34 37 47 52 64
---Program done, press RETURN---```