OFFSET
1,2
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = Sum_{k=1..n} k^3 * binomial(floor(n/k)+1,2).
G.f.: (1/(1-x)) * Sum_{k>=1} k^3 * x^k/(1 - x^k)^2.
a(n) ~ zeta(3) * n^4 / 4. - Vaclav Kotesovec, Aug 02 2022
MATHEMATICA
a[n_] := Sum[k * DivisorSigma[2, k], {k, 1, n}]; Array[a, 39] (* Amiram Eldar, Jul 28 2022 *)
PROG
(PARI) a(n) = sum(k=1, n, k*sigma(k, 2));
(PARI) a(n) = sum(k=1, n, k^3*binomial(n\k+1, 2));
(PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, k^3*x^k/(1-x^k)^2)/(1-x))
(Python)
from math import isqrt
def A356125(n): return (-((s:=isqrt(n))*(s+1))**3>>1) + sum(k*(q:=n//k)*(q+1)*(2*k**2+q*(q+1)) for k in range(1, s+1))>>2 # Chai Wah Wu, Oct 21 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Jul 27 2022
STATUS
approved