A237767 Integers whose product of digits is a nonzero cube.

1, 8, 11, 18, 24, 39, 42, 81, 88, 93, 111, 118, 124, 139, 142, 181, 188, 193, 214, 222, 241, 248, 284, 319, 333, 389, 391, 398, 412, 421, 428, 444, 469, 482, 496, 555, 649, 666, 694, 777, 811, 818, 824, 839, 842, 881, 888, 893, 913, 931

Offset: 1

Author

Derek Orr, Feb 12 2014

Keywords

• nonn
• base

Comments

No number with a 0 in it (A011540) can be in this sequence. If a number is in this sequence, then so is its reversal of digits (A004086) and other permutations of its digits. - Alonso del Arte, Feb 20 2014

Examples

			3*9*1 = 27 = 3^3, thus 391 is a member of this sequence.
3*9*8 = 216 = 6^3, thus 398 is a member of this sequence.
4*2*8 = 64 = 4^3, thus 428 is a member of this sequence.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A007954, A050626.

Programs

Maple

filter:= proc(n) local T;
  T:= Statistics:-Tally(convert(n,base,10),output=table);
  if assigned(T[0]) then return false fi;
  eval(T[2] + 2*T[4] + T[6] mod 3, T = [0$6]) = 0
  and eval(T[3] + T[6] + 2*T[9] mod 3, T = [0$9]) = 0
  and member(T[5] mod 3, [0,'T[5]'])
  and member(T[7] mod 3, [0,'T[7]'])
end proc:
select(filter, [$1..1000]); # Robert Israel, Jun 16 2025

Mathematica

pdcQ[n_]:=Module[{idn=IntegerDigits[n]},FreeQ[idn,0]&&IntegerQ[ Surd[ Times@@idn,3]]]; Select[Range[1000],pdcQ] (* Harvey P. Dale, Aug 25 2017 *)

PARI

s=[]; for(n=1, 1000, t=eval(Vec(Str(n))); d=prod(i=1, #t, t[i]); if(d>0 && ispower(d, 3), s=concat(s, n))); s \\ Colin Barker, Feb 17 2014

Python

def DigitProd(x):
  total = 1
  for i in str(x):
    total *= int(i)
  return total
def Cube(x):
  for n in range(1,10**3):
    if DigitProd(x) == n**3:
      return True
    if DigitProd(x) < n**3:
      return False
  return False
x = 1
while x < 1000:
  if Cube(x):
    print(x)
  x += 1

Python

from math import prod
from sympy import integer_nthroot
def ok(n): return (p:=prod(map(int, str(n)))) > 0 and integer_nthroot(p, 3)[1]
print([k for k in range(10**3) if ok(k)]) # Michael S. Branicky, Jun 16 2025

Formula

There are between 9^(k-6) and 9^k k-digit members of this sequence, so a(n) >> n^1.04 and in particular this sequence has density 0. - Charles R Greathouse IV, Feb 21 2014

Extensions

Name edited by Michel Marcus, Jun 16 2025