A356129 a(n) = Sum_{k=1..n} k * sigma_{n-1}(k).

1, 7, 41, 395, 4503, 68969, 1205345, 24831145, 574932340, 14936279962, 427782949566, 13426887958078, 457622797727840, 16842616079514468, 665489067204502336, 28102162931539093732, 1262904299189373463930, 60182778247311758955112

Offset: 1

Author

Seiichi Manyama, Jul 27 2022

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..386

Crossrefs

Cf. A356124, A356130, A356131.

Programs

Mathematica

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

PARI

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

PARI

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

Python

from math import isqrt
from sympy import bernoulli
def A356129(n): return ((s:=isqrt(n))*(s+1)*(bernoulli(n+1)-bernoulli(n+1,s+1))+sum(k**n*(n+1)*(q:=n//k)*(q+1)+(k*(bernoulli(n+1,q+1)-bernoulli(n+1))<<1) for k in range(1,s+1)))//(n+1)>>1 # Chai Wah Wu, Oct 24 2023

Formula

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

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

a(n) ~ c * n^n, where c = 1/(1 - 1/exp(1)) = A185393. - Vaclav Kotesovec, Aug 07 2022