A333844 -id:A333844 - cp's OEIS frontend

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

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, 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, 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

Offset: 1

Ilya Gutkovskiy, Apr 07 2020

Sum of cube roots of cube divisors of n.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Atul Dixit, Bibekananda Maji, and Akshaa Vatwani, Voronoi summation formula for the generalized divisor function sigma_z^k(n), arXiv:2303.09937 [math.NT], 2023, sigma(z=3,k=3,n).

Cf. A000203, A053150, A061704, A069290, A113061, A333844.

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
Table[DivisorSum[n, #^(1/3) &, IntegerQ[#^(1/3)] &], {n, 108}]
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 *)

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

Dirichlet g.f.: zeta(s) * zeta(3*s-1).
If n = Product (p_j^k_j) then a(n) = Product ((p_j^(floor(k_j/3) + 1) - 1)/(p_j - 1)).
Sum_{k=1..n} a(k) ~ Pi^2*n/6 + zeta(2/3)*n^(2/3)/2. - Vaclav Kotesovec, Dec 01 2020
a(n) = A000203(A053150(n)) (the sum of divisors of the cube root of largest cube dividing n). - Amiram Eldar, Sep 05 2023

A361794 Sum of the cubes d^3 of the divisors d satisfying d^2|n.

Original entry on oeis.org

1, 1, 1, 9, 1, 1, 1, 9, 28, 1, 1, 9, 1, 1, 1, 73, 1, 28, 1, 9, 1, 1, 1, 9, 126, 1, 28, 9, 1, 1, 1, 73, 1, 1, 1, 252, 1, 1, 1, 9, 1, 1, 1, 9, 28, 1, 1, 73, 344, 126, 1, 9, 1, 28, 1, 9, 1, 1, 1, 9, 1, 1, 28, 585, 1, 1, 1, 9, 1, 1, 1, 252, 1, 1, 126, 9, 1, 1, 1, 73

Offset: 1

R. J. Mathar, Mar 24 2023

The Mobius transform is 1, 0, 0, 8, 0, 0, 0, 0, 27, 0, 0, ... = n^(3/2)*A010052(n).

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=3,k=2,n).

Cf. A010052, A035316, A069290, A351308, A333843, A333844.

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,3,2),n=1..80) ;

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

a(n) = sumdiv(n, d, if (issquare(d), sqrtint(d)^3)); \\ Michel Marcus, Mar 24 2023

from math import prod
from sympy import factorint
def A361794(n): return prod((p**(3*(e>>1)+3)-1)//(p**3-1) for p, e in factorint(n).items()) # Chai Wah Wu, Mar 24 2023

a(n) = Sum_{d^2|n} d^3.
Multiplicative with a(p^e) = (p^(3*(floor(e/2) + 1)) - 1)/(p^3 - 1). - Amiram Eldar, Mar 24 2023
G.f.: Sum_{k>=1} k^3 * x^(k^2) / (1 - x^(k^2)). - Ilya Gutkovskiy, Jun 05 2024

cp's OEIS Frontend

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

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