A361793 - OEIS

%I #30 Jun 26 2024 04:32:50

%S 1,1,1,1,1,1,1,5,1,1,1,1,1,1,1,5,1,1,1,1,1,1,1,5,1,1,10,1,1,1,1,5,1,1,

%T 1,1,1,1,1,5,1,1,1,1,1,1,1,5,1,1,1,1,1,10,1,5,1,1,1,1,1,1,1,21,1,1,1,

%U 1,1,1,1,5,1,1,1,1,1,1,1,5

%N Sum of the squares d^2 of the divisors d satisfying d^3|n.

%C The Mobius transform is 1, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, ... = n^(2/3)*A010057(n).

%H Winston de Greef, <a href="/A361793/b361793.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=2,k=3,n).

%F a(n) = Sum_{d^3|n} d^2.

%F Multiplicative with a(p^e) = (p^(2*(floor(e/3) + 1)) - 1)/(p^2 - 1). - _Amiram Eldar_, Mar 24 2023

%F G.f.: Sum_{k>=1} k^2 * x^(k^3) / (1 - x^(k^3)). - _Ilya Gutkovskiy_, Jun 05 2024

%F From _Vaclav Kotesovec_, Jun 26 2024: (Start)

%F Dirichlet g.f.: zeta(s)*zeta(3*s-2).

%F Sum_{k=1..n} a(k) ~ n*(log(n) + 4*gamma - 1)/3, where gamma is the Euler-Mascheroni constant A001620. (End)

%p gsigma := proc(n,z,k)

%p     local a,d ;

%p     a := 0 ;

%p     for d in numtheory[divisors](n) do

%p         if modp(n,d^k) = 0 then

%p             a := a+d^z ;

%p         end if ;

%p     end do:

%p     a ;

%p end proc:

%p seq( gsigma(n,2,3),n=1..80) ;

%t f[p_, e_] := (p^(2*(Floor[e/3] + 1)) - 1)/(p^2 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Mar 24 2023 *)

%o (PARI) a(n) = sumdiv(n, d, if (ispower(d, 3), sqrtnint(d, 3)^2)); \\ _Michel Marcus_, Mar 24 2023

%o (PARI) for(n=1, 100, print1(direuler(p=2, n, (1/((1-X)*(1-p^2*X^3))))[n], ", ")) \\ _Vaclav Kotesovec_, Jun 26 2024

%o (Python)

%o from math import prod

%o from sympy import factorint

%o def A361793(n): return prod((p**(e//3+1<<1)-1)//(p**2-1) for p, e in factorint(n).items()) # _Chai Wah Wu_, Mar 24 2023

%Y Cf. A035316, A069290, A333843.

%K nonn,mult,easy

%O 1,8

%A _R. J. Mathar_, Mar 24 2023