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A364269 a(n) = Sum_{k=1..n} k^3*sigma_2(k), where sigma_2 is A001157.

%I A364269 #32 Oct 22 2023 00:45:11
%S A364269 1,41,311,1655,4905,15705,32855,76375,142714,272714,435096,797976,
%T A364269 1171466,1857466,2734966,4131702,5556472,8210032,10692990,15060990,
%U A364269 19691490,26186770,32635280,44385680,54557555,69497155,85637215,108686815,129222353,164322353
%N A364269 a(n) = Sum_{k=1..n} k^3*sigma_2(k), where sigma_2 is A001157.
%H A364269 Wikipedia, <a href="https://en.wikipedia.org/wiki/Faulhaber%27s_formula">Faulhaber's formula</a>.
%F A364269 a(n) = Sum_{k=1..n} k^5 * A000537(floor(n/k)).
%F A364269 a(n) ~ (zeta(3)/6) * n^6. - _Amiram Eldar_, Oct 20 2023
%t A364269 Accumulate[Table[n^3*DivisorSigma[2, n], {n, 1, 30}]] (* _Amiram Eldar_, Oct 20 2023 *)
%o A364269 (PARI) f(n, m) = (subst(bernpol(m+1, x), x, n+1)-subst(bernpol(m+1, x), x, 0))/(m+1);
%o A364269 a(n, s=3, t=2) = sum(k=1, n, k^(s+t)*f(n\k, s));
%o A364269 (Python)
%o A364269 def A364269(n): return sum(k*(k**2*(m:=n//k)*(m+1)>>1)**2 for k in range(1,n+1)) # _Chai Wah Wu_, Oct 20 2023
%o A364269 (Python)
%o A364269 from math import isqrt
%o A364269 def A364269(n): return ((((s:=isqrt(n))*(s+1))**4*(1-s*(s+1<<1))>>2) + sum(((q:=n//k)*(q+1))**2*k**3*(3*k**2+(q*(q+1<<1)-1)) for k in range(1,s+1)))//12 # _Chai Wah Wu_, Oct 21 2023
%Y A364269 Cf. A356125, A364268.
%Y A364269 Cf. A000537, A001157.
%K A364269 nonn,easy
%O A364269 1,2
%A A364269 _Seiichi Manyama_, Oct 20 2023