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