cp's OEIS Frontend

A356243 a(n) = Sum_{k=1..n} k^2 * sigma_{n-2}(k).

%I A356243 #16 Oct 21 2023 19:38:55
%S A356243 1,9,49,447,4607,71009,1210855,24980627,575624572,14958422046,
%T A356243 427890493960,13431874937840,457651929853662,16844143705998554,
%U A356243 665499756005678382,28102799297908820326,1262909308355648335240,60183118566605371095996
%N A356243 a(n) = Sum_{k=1..n} k^2 * sigma_{n-2}(k).
%F A356243 a(n) = Sum_{k=1..n} k^n * Sum_{j=1..floor(n/k)} j^2 = Sum_{k=1..n} k^n * A000330(floor(n/k)).
%F A356243 a(n) = [x^n] (1/(1-x)) * Sum_{k>=1} k^n * x^k * (1 + x^k)/(1 - x^k)^3.
%t A356243 a[n_] := Sum[k^2 * DivisorSigma[n - 2, k], {k, 1, n}]; Array[a, 18] (* _Amiram Eldar_, Jul 30 2022 *)
%o A356243 (PARI) a(n) = sum(k=1, n, k^2*sigma(k, n-2));
%o A356243 (PARI) a(n) = sum(k=1, n, k^n*sum(j=1, n\k, j^2));
%o A356243 (Python)
%o A356243 from math import isqrt
%o A356243 from sympy import bernoulli
%o A356243 def A356243(n): return (((s:=isqrt(n))*(s+1)*(2*s+1))*((b:=bernoulli(n+1))-bernoulli(n+1, s+1)) + sum(k**n*(n+1)*((q:=n//k)*(q+1)*(2*q+1))+6*k**2*(bernoulli(n+1,q+1)-b) for k in range(1,s+1)))//(n+1)//6 # _Chai Wah Wu_, Oct 21 2023
%Y A356243 Cf. A000330, A319194, A356129, A356239.
%K A356243 nonn
%O A356243 1,2
%A A356243 _Seiichi Manyama_, Jul 30 2022