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

%I A048944 #27 Feb 16 2025 08:32:40
%S A048944 1,4,8,9,12,18,20,25,27,28,32,36,44,45,49,50,52,60,63,64,68,72,75,76,
%T A048944 84,90,92,96,98,99,100,108,116,117,121,124,125,126,132,140,144,147,
%U A048944 148,150,153,156,160,164,169,171,172,175,180,188,196,198,200,204,207
%N A048944 Numbers k such that the product of divisors of k is a cube.
%C A048944 From _Robert Israel_, Jun 30 2014: (Start)
%C A048944 n is in the sequence iff either
%C A048944 1) for at least one prime p dividing n, the p-adic order of n is congruent to 2 mod 3, or
%C A048944 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)
%C A048944 The asymptotic density of this sequence is 1 - zeta(3)/zeta(2) = 0.2692370305... . - _Amiram Eldar_, Jul 01 2022
%H A048944 Robert Israel, <a href="/A048944/b048944.txt">Table of n, a(n) for n = 1..10000</a>
%H A048944 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DivisorProduct.html">Divisor Product</a>.
%p A048944 filter:=  proc(n) local F;
%p A048944       F:= ifactors(n)[2];
%p A048944       F:= convert(map(t -> t[2] mod 3, F),set);
%p A048944       has(F,2) or F = {0} or F = {};
%p A048944 end proc:
%p A048944 select(filter, [$1..1000]); # _Robert Israel_, Jun 30 2014
%t A048944 Select[Range[250],IntegerQ[Surd[Times@@Divisors[#],3]]&] (* _Harvey P. Dale_, Feb 05 2019 *)
%t A048944 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 *)
%o A048944 (PARI) is(n)=ispower(n,3) || #select(e->e%3==2, factor(n)[,2]) \\ _Charles R Greathouse IV_, Sep 18 2015
%Y A048944 Disjoint union of A000578 and A059269.
%Y A048944 Cf. A007955.
%K A048944 nonn
%O A048944 1,2
%A A048944 _Eric W. Weisstein_