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A333843 Expansion of g.f.: Sum_{k>=1} k * x^(k^3) / (1 - x^(k^3)).

%I A333843 #23 Sep 06 2023 01:15:17
%S A333843 1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,3,1,1,4,1,1,1,1,3,1,1,
%T A333843 1,1,1,1,1,3,1,1,1,1,1,1,1,3,1,1,1,1,1,4,1,3,1,1,1,1,1,1,1,7,1,1,1,1,
%U A333843 1,1,1,3,1,1,1,1,1,1,1,3,4,1,1,1,1,1,1,3,1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,3,1,1,1,4
%N A333843 Expansion of g.f.: Sum_{k>=1} k * x^(k^3) / (1 - x^(k^3)).
%C A333843 Sum of cube roots of cube divisors of n.
%H A333843 Amiram Eldar, <a href="/A333843/b333843.txt">Table of n, a(n) for n = 1..10000</a>
%H A333843 Atul Dixit, Bibekananda Maji, and Akshaa Vatwani, <a href="https://arxiv.org/abs/2303.09937">Voronoi summation formula for the generalized divisor function sigma_z^k(n)</a>, arXiv:2303.09937 [math.NT], 2023, sigma(z=3,k=3,n).
%F A333843 Dirichlet g.f.: zeta(s) * zeta(3*s-1).
%F A333843 If n = Product (p_j^k_j) then a(n) = Product ((p_j^(floor(k_j/3) + 1) - 1)/(p_j - 1)).
%F A333843 Sum_{k=1..n} a(k) ~ Pi^2*n/6 + zeta(2/3)*n^(2/3)/2. - _Vaclav Kotesovec_, Dec 01 2020
%F A333843 a(n) = A000203(A053150(n)) (the sum of divisors of the cube root of largest cube dividing n). - _Amiram Eldar_, Sep 05 2023
%t A333843 nmax = 108; CoefficientList[Series[Sum[k x^(k^3)/(1 - x^(k^3)), {k, 1, Floor[nmax^(1/3)] + 1}], {x, 0, nmax}], x] // Rest
%t A333843 Table[DivisorSum[n, #^(1/3) &, IntegerQ[#^(1/3)] &], {n, 108}]
%t A333843 f[p_, e_] := (p^(Floor[e/3] + 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Dec 01 2020 *)
%o A333843 (PARI) a(n) = {my(f = factor(n)); prod(i=1, #f~, (f[i,1]^(f[i,2]\3 + 1) - 1)/(f[i,1] - 1));} \\ _Amiram Eldar_, Sep 05 2023
%Y A333843 Cf. A000203, A053150, A061704, A069290, A113061, A333844.
%K A333843 nonn,easy,mult
%O A333843 1,8
%A A333843 _Ilya Gutkovskiy_, Apr 07 2020