A333844 - OEIS

%I #18 Jan 24 2024 12:19:06

%S 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,1,1,

%T 1,1,1,1,1,1,1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,1,1,1,1,

%U 1,1,1,1,1,1,1,1,1,1,1,3,4,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3

%N G.f.: Sum_{k>=1} k * x^(k^4) / (1 - x^(k^4)).

%C Sum of 4th roots of 4th powers dividing n.

%H Amiram Eldar, <a href="/A333844/b333844.txt">Table of n, a(n) for n = 1..10000</a>

%H A. Dixit, B. Maji, and A. 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=1,k=4,n).

%F Dirichlet g.f.: zeta(s) * zeta(4*s-1).

%F If n = Product (p_j^k_j) then a(n) = Product ((p_j^(floor(k_j/4) + 1) - 1)/(p_j - 1)).

%F Sum_{k=1..n} a(k) ~ zeta(3)*n + zeta(1/2)*sqrt(n)/2. - _Vaclav Kotesovec_, Dec 01 2020

%t nmax = 112; CoefficientList[Series[Sum[k x^(k^4)/(1 - x^(k^4)), {k, 1, Floor[nmax^(1/4)] + 1}], {x, 0, nmax}], x] // Rest

%t Table[DivisorSum[n, #^(1/4) &, IntegerQ[#^(1/4)] &], {n, 112}]

%t f[p_, e_] := (p^(Floor[e/4] + 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Dec 01 2020 *)

%Y Cf. A000203, A053164, A063775, A069290, A300909, A333843.

%K nonn,mult

%O 1,16

%A _Ilya Gutkovskiy_, Apr 07 2020