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a(n) = Sum_{k=1..n} k^3*sigma(k), where sigma is A000203.
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%I #34 Oct 22 2023 00:47:05

%S 1,25,133,581,1331,3923,6667,14347,23824,41824,57796,106180,136938,

%T 202794,283794,410770,499204,726652,863832,1199832,1496184,1879512,

%U 2171520,3000960,3485335,4223527,5010847,6240159,6971829,8915829,9869141,11933525,13658501

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

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Faulhaber%27s_formula">Faulhaber's formula</a>.

%F a(n) = Sum_{k=1..n} k^4 * A000537(floor(n/k)).

%F a(n) ~ (zeta(2)/5) * n^5. - _Amiram Eldar_, Oct 20 2023

%t Accumulate[Table[n^3*DivisorSigma[1, n], {n, 1, 33}]] (* _Amiram Eldar_, Oct 20 2023 *)

%o (PARI) f(n, m) = (subst(bernpol(m+1, x), x, n+1)-subst(bernpol(m+1, x), x, 0))/(m+1);

%o a(n, s=3, t=1) = sum(k=1, n, k^(s+t)*f(n\k, s));

%o (Python)

%o 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 (Python)

%o from math import isqrt

%o 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 Partial sums of A282211.

%Y Cf. A319086, A320895, A356125, A364269.

%Y Cf. A000203, A000537, A143128.

%K nonn,easy

%O 1,2

%A _Seiichi Manyama_, Oct 20 2023