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A237767 Integers whose product of digits is a nonzero cube.

%I A237767 #26 Jun 16 2025 15:32:57
%S A237767 1,8,11,18,24,39,42,81,88,93,111,118,124,139,142,181,188,193,214,222,
%T A237767 241,248,284,319,333,389,391,398,412,421,428,444,469,482,496,555,649,
%U A237767 666,694,777,811,818,824,839,842,881,888,893,913,931
%N A237767 Integers whose product of digits is a nonzero cube.
%C A237767 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
%H A237767 Harvey P. Dale, <a href="/A237767/b237767.txt">Table of n, a(n) for n = 1..1000</a>
%F A237767 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
%e A237767 3*9*1 = 27 = 3^3, thus 391 is a member of this sequence.
%e A237767 3*9*8 = 216 = 6^3, thus 398 is a member of this sequence.
%e A237767 4*2*8 = 64 = 4^3, thus 428 is a member of this sequence.
%p A237767 filter:= proc(n) local T;
%p A237767   T:= Statistics:-Tally(convert(n,base,10),output=table);
%p A237767   if assigned(T[0]) then return false fi;
%p A237767   eval(T[2] + 2*T[4] + T[6] mod 3, T = [0$6]) = 0
%p A237767   and eval(T[3] + T[6] + 2*T[9] mod 3, T = [0$9]) = 0
%p A237767   and member(T[5] mod 3, [0,'T[5]'])
%p A237767   and member(T[7] mod 3, [0,'T[7]'])
%p A237767 end proc:
%p A237767 select(filter, [$1..1000]); # _Robert Israel_, Jun 16 2025
%t A237767 pdcQ[n_]:=Module[{idn=IntegerDigits[n]},FreeQ[idn,0]&&IntegerQ[ Surd[ Times@@idn,3]]]; Select[Range[1000],pdcQ] (* _Harvey P. Dale_, Aug 25 2017 *)
%o A237767 (Python)
%o A237767 def DigitProd(x):
%o A237767   total = 1
%o A237767   for i in str(x):
%o A237767     total *= int(i)
%o A237767   return total
%o A237767 def Cube(x):
%o A237767   for n in range(1,10**3):
%o A237767     if DigitProd(x) == n**3:
%o A237767       return True
%o A237767     if DigitProd(x) < n**3:
%o A237767       return False
%o A237767   return False
%o A237767 x = 1
%o A237767 while x < 1000:
%o A237767   if Cube(x):
%o A237767     print(x)
%o A237767   x += 1
%o A237767 (Python)
%o A237767 from math import prod
%o A237767 from sympy import integer_nthroot
%o A237767 def ok(n): return (p:=prod(map(int, str(n)))) > 0 and integer_nthroot(p, 3)[1]
%o A237767 print([k for k in range(10**3) if ok(k)]) # _Michael S. Branicky_, Jun 16 2025
%o A237767 (PARI)
%o A237767 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
%Y A237767 Cf. A007954, A050626.
%K A237767 nonn,base
%O A237767 1,2
%A A237767 _Derek Orr_, Feb 12 2014
%E A237767 Name edited by _Michel Marcus_, Jun 16 2025