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

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