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