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A364194 a(n) = Sum_{k=1..n} k^3*sigma(k), where sigma is A000203.

%I A364194 #34 Oct 22 2023 00:47:05
%S A364194 1,25,133,581,1331,3923,6667,14347,23824,41824,57796,106180,136938,
%T A364194 202794,283794,410770,499204,726652,863832,1199832,1496184,1879512,
%U A364194 2171520,3000960,3485335,4223527,5010847,6240159,6971829,8915829,9869141,11933525,13658501
%N A364194 a(n) = Sum_{k=1..n} k^3*sigma(k), where sigma is A000203.
%H A364194 Wikipedia, <a href="https://en.wikipedia.org/wiki/Faulhaber%27s_formula">Faulhaber's formula</a>.
%F A364194 a(n) = Sum_{k=1..n} k^4 * A000537(floor(n/k)).
%F A364194 a(n) ~ (zeta(2)/5) * n^5. - _Amiram Eldar_, Oct 20 2023
%t A364194 Accumulate[Table[n^3*DivisorSigma[1, n], {n, 1, 33}]] (* _Amiram Eldar_, Oct 20 2023 *)
%o A364194 (PARI) f(n, m) = (subst(bernpol(m+1, x), x, n+1)-subst(bernpol(m+1, x), x, 0))/(m+1);
%o A364194 a(n, s=3, t=1) = sum(k=1, n, k^(s+t)*f(n\k, s));
%o A364194 (Python)
%o A364194 def A364194(n): return sum((k**2*(m:=n//k)*(m+1)>>1)**2 for k in range(1,n+1)) # _Chai Wah Wu_, Oct 20 2023
%o A364194 (Python)
%o A364194 from math import isqrt
%o A364194 def A364194(n): return ((((s:=isqrt(n))*(s + 1))**3*(2*s+1)*(1-3*s*(s+1))>>1) + sum((q:=n//k)*(q+1)*k**3*(q*(15*k+q*(15*k+12*q+18)+2)-2) for k in range(1,s+1)))//60 # _Chai Wah Wu_, Oct 21 2023
%Y A364194 Partial sums of A282211.
%Y A364194 Cf. A319086, A320895, A356125, A364269.
%Y A364194 Cf. A000203, A000537, A143128.
%K A364194 nonn,easy
%O A364194 1,2
%A A364194 _Seiichi Manyama_, Oct 20 2023