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A294072 Number of noncube divisors of n^3.

%I A294072 #13 Jan 16 2020 00:59:47
%S A294072 0,2,2,4,2,12,2,6,4,12,2,22,2,12,12,8,2,22,2,22,12,12,2,32,4,12,6,22,
%T A294072 2,56,2,10,12,12,12,40,2,12,12,32,2,56,2,22,22,12,2,42,4,22,12,22,2,
%U A294072 32,12,32,12,12,2,100,2,12,22,12,12,56,2,22,12,56,2,58,2,12,22,22,12,56,2,42,8,12,2,100,12
%N A294072 Number of noncube divisors of n^3.
%C A294072 All terms are even.  a(n)=2 if and only if n is prime. - _Robert Israel_, Jan 16 2020
%H A294072 Robert Israel, <a href="/A294072/b294072.txt">Table of n, a(n) for n = 1..10000</a>
%H A294072 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A294072 G.f.: Sum_{k>=1} (3^omega(k) - 1)*x^k/(1 - x^k), where omega(k) is the number of distinct primes dividing k (A001221).
%F A294072 a(n) = [x^(n^3)] Sum_{k>=1} x^A007412(k)/(1 - x^A007412(k)).
%F A294072 a(n) = A291208(A000578(n)).
%F A294072 a(n) = A048785(n) - A000005(n).
%e A294072 a(4) = 4 because 4^3 = 64 has 7 divisors {1, 2, 4, 8, 16, 32, 64} among which 4 divisors {2, 4, 16, 32} are noncubes.
%p A294072 f:= proc(n) local F;
%p A294072   F:= map(t -> t[2],ifactors(n)[2]);
%p A294072   mul(1+3*t,t=F) - mul(1+t,t=F)
%p A294072 end proc:
%p A294072 map(f, [$1..100]; # _Robert Israel_, Jan 16 2020
%t A294072 nmax = 85; Rest[CoefficientList[Series[Sum[(3^PrimeNu[k] - 1) x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
%t A294072 a[n_] := Length[Select[Divisors[n], ! IntegerQ[#^(1/3)] &]]; Table[a[n^3], {n, 1, 85}]
%t A294072 Table[DivisorSigma[0, n^3] - DivisorSigma[0, n], {n, 1, 85}]
%Y A294072 Cf. A000005, A000578, A001221, A007412, A048785, A055205, A061704, A291208.
%K A294072 nonn
%O A294072 1,2
%A A294072 _Ilya Gutkovskiy_, Feb 07 2018