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A350128 a(n) = Sum_{k=1..n} k^n * floor(n/k)^2.

%I A350128 #22 Oct 21 2023 19:58:14
%S A350128 1,8,44,417,4545,69905,1207937,24904806,575256641,14947281595,
%T A350128 427836523971,13429362462839,457637290140469,16843379604615375,
%U A350128 665494379869134005,28102480944522059434,1262906802939553227382,60182948301301262753877
%N A350128 a(n) = Sum_{k=1..n} k^n * floor(n/k)^2.
%H A350128 Seiichi Manyama, <a href="/A350128/b350128.txt">Table of n, a(n) for n = 1..386</a>
%F A350128 a(n) = Sum_{k=1..n} 2 * k * sigma_{n-1}(k) - sigma_{n}(k).
%F A350128 a(n) ~ n^n / (1 - exp(-1)). - _Vaclav Kotesovec_, Dec 16 2021
%t A350128 Table[Sum[k^n Floor[n/k]^2,{k,n}],{n,20}] (* _Harvey P. Dale_, Feb 11 2022 *)
%o A350128 (PARI) a(n) = sum(k=1, n, k^n*(n\k)^2);
%o A350128 (PARI) a(n) = sum(k=1, n, 2*k*sigma(k, n-1)-sigma(k, n));
%o A350128 (Python)
%o A350128 from math import isqrt
%o A350128 from sympy import bernoulli
%o A350128 def A350128(n): return (((s:=isqrt(n))+1)*(1-s)*(bernoulli(n+1,s+1)-(b:=bernoulli(n+1)))+sum(k**n*(n+1)*(((q:=n//k)+1)*(q-1))+(1-2*k)*(b-bernoulli(n+1,q+1)) for k in range(1,s+1)))//(n+1) # _Chai Wah Wu_, Oct 21 2023
%Y A350128 Cf. A222548, A319194, A350107, A350125.
%K A350128 nonn
%O A350128 1,2
%A A350128 _Seiichi Manyama_, Dec 15 2021