A048944 - OEIS

%I #26 Jul 01 2022 05:27:00

%S 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 84,90,92,96,98,99,100,108,116,117,121,124,125,126,132,140,144,147,

%U 148,150,153,156,160,164,169,171,172,175,180,188,196,198,200,204,207

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

%C From _Robert Israel_, Jun 30 2014: (Start)

%C n is in the sequence iff either

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

%C 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 The asymptotic density of this sequence is 1 - zeta(3)/zeta(2) = 0.2692370305... . - _Amiram Eldar_, Jul 01 2022

%H Robert Israel, <a href="/A048944/b048944.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DivisorProduct.html">Divisor Product</a>.

%p filter:=  proc(n) local F;

%p       F:= ifactors(n)[2];

%p       F:= convert(map(t -> t[2] mod 3, F),set);

%p       has(F,2) or F = {0} or F = {};

%p end proc:

%p select(filter, [$1..1000]); # _Robert Israel_, Jun 30 2014

%t Select[Range[250],IntegerQ[Surd[Times@@Divisors[#],3]]&] (* _Harvey P. Dale_, Feb 05 2019 *)

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

%Y Disjoint union of A000578 and A059269.

%Y Cf. A007955.

%K nonn

%O 1,2

%A _Eric W. Weisstein_