Sum of square and cube digits of an integer are primes
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.
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.
- Task
- 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 #
PR read "primes.incl.a68" PR
[]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.
ADD DIGITS(D) TO SUM.
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
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
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
Haskell
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
jq
Also works with gojq and fq
Preliminaries
def is_prime:
. as $n
| if ($n < 2) then false
elif ($n % 2 == 0) then $n == 2
elif ($n % 3 == 0) then $n == 3
elif ($n % 5 == 0) then $n == 5
elif ($n % 7 == 0) then $n == 7
elif ($n % 11 == 0) then $n == 11
elif ($n % 13 == 0) then $n == 13
elif ($n % 17 == 0) then $n == 17
elif ($n % 19 == 0) then $n == 19
else 23
| until( (. * .) > $n or ($n % . == 0); .+2)
| . * . > $n
end;
def digitSum: tostring | explode | map([.] | implode | tonumber) | add;The Task
range(1;100)
| (.*.) as $sq
| select( ($sq | digitSum | is_prime) and ($sq * . | digitSum | is_prime ) )- 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]
MAD
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
OCaml
let is_prime n =
let rec test x =
let q = n / x in x > q || x * q <> n && n mod (x + 2) <> 0 && test (x + 6)
in if n < 5 then n lor 1 = 3 else n land 1 <> 0 && n mod 3 <> 0 && test 5
let rec digit_sum n =
if n < 10 then n else n mod 10 + digit_sum (n / 10)
let is_square_and_cube_digit_sum_prime n =
is_prime (digit_sum (n * n)) && is_prime (digit_sum (n * n * n))
let () =
Seq.ints 1 |> Seq.take_while ((>) 100)
|> Seq.filter is_square_and_cube_digit_sum_prime
|> Seq.iter (Printf.printf " %u") |> print_newline
- Output:
16 17 25 28 34 37 47 52 64
Perl
#!/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
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
// 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---