A361793 Sum of the squares d^2 of the divisors d satisfying d^3|n.

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, 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, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 5

Offset

1,8

Comments

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

Links

Winston de Greef, Table of n, a(n) for n = 1..10000

A. Dixit, B. Maji, and A. Vatwani, Voronoi summation formula for the generalized divisor function sigma_z^k(n), arXiv:2303.09937 [math.NT], 2023, sigma(z=2,k=3,n).

Formula

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

Multiplicative with a(p^e) = (p^(2*(floor(e/3) + 1)) - 1)/(p^2 - 1). - Amiram Eldar, Mar 24 2023

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

From Vaclav Kotesovec, Jun 26 2024: (Start)

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

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

Maple

gsigma := proc(n, z, k)
    local a, d ;
    a := 0 ;
    for d in numtheory[divisors](n) do
        if modp(n, d^k) = 0 then
            a := a+d^z ;
        end if ;
    end do:
    a ;
end proc:
seq( gsigma(n, 2, 3), n=1..80) ;

Mathematica

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 *)

Prog

(PARI) a(n) = sumdiv(n, d, if (ispower(d, 3), sqrtnint(d, 3)^2)); \\ Michel Marcus, Mar 24 2023
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1/((1-X)*(1-p^2*X^3))))[n], ", ")) \\ Vaclav Kotesovec, Jun 26 2024
(Python)
from math import prod
from sympy import factorint
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

Crossrefs

Cf. A035316, A069290, A333843.

Sequence in context: A319099 A098087 A354987 * A257099 A348948 A291578

Adjacent sequences: A361790 A361791 A361792 * A361794 A361795 A361796

Keyword

nonn,mult,easy

Author

R. J. Mathar, Mar 24 2023

Status

approved