A048944 Numbers k such that the product of divisors of k is a cube.

1, 4, 8, 9, 12, 18, 20, 25, 27, 28, 32, 36, 44, 45, 49, 50, 52, 60, 63, 64, 68, 72, 75, 76, 84, 90, 92, 96, 98, 99, 100, 108, 116, 117, 121, 124, 125, 126, 132, 140, 144, 147, 148, 150, 153, 156, 160, 164, 169, 171, 172, 175, 180, 188, 196, 198, 200, 204, 207

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn

Comments

From Robert Israel, Jun 30 2014: (Start)

n is in the sequence iff either

1) for at least one prime p dividing n, the p-adic order of n is congruent to 2 mod 3, or

2) for all primes p dividing n, the p-adic order of n is congruent to 0 mod 3 (and thus n is a cube). (End)

The asymptotic density of this sequence is 1 - zeta(3)/zeta(2) = 0.2692370305... . - Amiram Eldar, Jul 01 2022

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Divisor Product.

Crossrefs

Disjoint union of A000578 and A059269.

Cf. A007955.

Programs

Maple

filter:=  proc(n) local F;
      F:= ifactors(n)[2];
      F:= convert(map(t -> t[2] mod 3, F),set);
      has(F,2) or F = {0} or F = {};
end proc:
select(filter, [$1..1000]); # Robert Israel, Jun 30 2014

Mathematica

Select[Range[250],IntegerQ[Surd[Times@@Divisors[#],3]]&] (* Harvey P. Dale, Feb 05 2019 *)
q[n_] := AnyTrue[FactorInteger[n][[;; , 2]], Mod[#, 3] == 2 &]; m = 6; Union[Range[m]^3, Select[Range[m^3], q]] (* Amiram Eldar, Jul 01 2022 *)

PARI

is(n)=ispower(n,3) || #select(e->e%3==2, factor(n)[,2]) \\ Charles R Greathouse IV, Sep 18 2015