%I #22 Oct 24 2023 15:43:37
%S 1,7,41,395,4503,68969,1205345,24831145,574932340,14936279962,
%T 427782949566,13426887958078,457622797727840,16842616079514468,
%U 665489067204502336,28102162931539093732,1262904299189373463930,60182778247311758955112
%N a(n) = Sum_{k=1..n} k * sigma_{n-1}(k).
%H Seiichi Manyama, <a href="/A356129/b356129.txt">Table of n, a(n) for n = 1..386</a>
%F a(n) = Sum_{k=1..n} k^n * binomial(floor(n/k)+1,2).
%F a(n) = [x^n] (1/(1-x)) * Sum_{k>=1} k^n * x^k/(1 - x^k)^2.
%F a(n) ~ c * n^n, where c = 1/(1 - 1/exp(1)) = A185393. - _Vaclav Kotesovec_, Aug 07 2022
%t a[n_] := Sum[k * DivisorSigma[n - 1, k], {k, 1, n}]; Array[a, 18] (* _Amiram Eldar_, Jul 28 2022 *)
%o (PARI) a(n) = sum(k=1, n, k*sigma(k, n-1));
%o (PARI) a(n) = sum(k=1, n, k^n*binomial(n\k+1, 2));
%o (Python)
%o from math import isqrt
%o from sympy import bernoulli
%o 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
%Y Cf. A356124, A356130, A356131.
%K nonn
%O 1,2
%A _Seiichi Manyama_, Jul 27 2022