A356128 a(n) = Sum_{k=1..n} k * sigma_n(k).

1, 11, 103, 1373, 20657, 381795, 7921825, 187452793, 4916743582, 142471278944, 4506381463150, 154747691135574, 5729252807696052, 227595085199164036, 9654855890695727316, 435664037303036699736, 20836069678062430493950, 1052867409176853099312712

Offset: 1

Author

Seiichi Manyama, Jul 27 2022

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..385

Crossrefs

Cf. A356046, A356124.

Programs

Mathematica

a[n_] := Sum[k * DivisorSigma[n, k], {k, 1, n}]; Array[a, 18] (* Amiram Eldar, Jul 28 2022 *)

PARI

a(n) = sum(k=1, n, k*sigma(k, n));

PARI

a(n) = sum(k=1, n, k^(n+1)*binomial(n\k+1, 2));

Python

from math import isqrt
from sympy import bernoulli
def A356128(n): return ((s:=isqrt(n))*(s+1)*(bernoulli(n+2)-bernoulli(n+2,s+1))+sum(k**(n+1)*(n+2)*(q:=n//k)*(q+1)+(k*(bernoulli(n+2,q+1)-bernoulli(n+2))<<1) for k in range(1,s+1)))//(n+2)>>1 # Chai Wah Wu, Oct 24 2023

Formula

a(n) = Sum_{k=1..n} k^(n+1) * binomial(floor(n/k)+1,2).

a(n) = [x^n] (1/(1-x)) * Sum_{k>=1} k^(n+1) * x^k/(1 - x^k)^2.