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

%I A356128 #21 Oct 24 2023 15:43:33
%S A356128 1,11,103,1373,20657,381795,7921825,187452793,4916743582,142471278944,
%T A356128 4506381463150,154747691135574,5729252807696052,227595085199164036,
%U A356128 9654855890695727316,435664037303036699736,20836069678062430493950,1052867409176853099312712
%N A356128 a(n) = Sum_{k=1..n} k * sigma_n(k).
%H A356128 Seiichi Manyama, <a href="/A356128/b356128.txt">Table of n, a(n) for n = 1..385</a>
%F A356128 a(n) = Sum_{k=1..n} k^(n+1) * binomial(floor(n/k)+1,2).
%F A356128 a(n) = [x^n] (1/(1-x)) * Sum_{k>=1} k^(n+1) * x^k/(1 - x^k)^2.
%t A356128 a[n_] := Sum[k * DivisorSigma[n, k], {k, 1, n}]; Array[a, 18] (* _Amiram Eldar_, Jul 28 2022 *)
%o A356128 (PARI) a(n) = sum(k=1, n, k*sigma(k, n));
%o A356128 (PARI) a(n) = sum(k=1, n, k^(n+1)*binomial(n\k+1, 2));
%o A356128 (Python)
%o A356128 from math import isqrt
%o A356128 from sympy import bernoulli
%o A356128 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
%Y A356128 Cf. A356046, A356124.
%K A356128 nonn
%O A356128 1,2
%A A356128 _Seiichi Manyama_, Jul 27 2022