cp's OEIS Frontend

A350123 a(n) = Sum_{k=1..n} k^2 * floor(n/k)^2.

%I A350123 #29 Oct 24 2023 17:58:10
%S A350123 1,8,22,57,91,185,247,402,545,775,917,1379,1573,1995,2455,3106,3428,
%T A350123 4377,4775,5909,6753,7727,8301,10331,11230,12564,13904,15990,16888,
%U A350123 19908,20930,23597,25545,27767,29827,34468,35910,38660,41328,46318,48080,53644,55578
%N A350123 a(n) = Sum_{k=1..n} k^2 * floor(n/k)^2.
%H A350123 Seiichi Manyama, <a href="/A350123/b350123.txt">Table of n, a(n) for n = 1..10000</a>
%F A350123 a(n) = Sum_{k=1..n} k^2 * Sum_{d|k} (2*d - 1)/d^2 = Sum_{k=1..n} 2 * k * sigma(k) - sigma_2(k) = 2 * A143128(n) - A064602(n).
%F A350123 G.f.: (1/(1 - x)) * Sum_{k>=1} (2*k - 1) * x^k * (1 + x^k)/(1 - x^k)^3.
%F A350123 a(n) ~ n^3 * (Pi^2/9 - zeta(3)/3). - _Vaclav Kotesovec_, Dec 16 2021
%t A350123 Accumulate[Table[2*k*DivisorSigma[1, k] - DivisorSigma[2, k], {k, 1, 50}]] (* _Vaclav Kotesovec_, Dec 16 2021 *)
%o A350123 (PARI) a(n) = sum(k=1, n, k^2*(n\k)^2);
%o A350123 (PARI) a(n) = sum(k=1, n, k^2*sumdiv(k, d, (2*d-1)/d^2));
%o A350123 (PARI) a(n) = sum(k=1, n, 2*k*sigma(k)-sigma(k, 2));
%o A350123 (PARI) my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, (2*k-1)*x^k*(1+x^k)/(1-x^k)^3)/(1-x))
%o A350123 (Python)
%o A350123 from math import isqrt
%o A350123 def A350123(n): return (-(s:=isqrt(n))**3*(s+1)*((s<<1)+1)+sum((q:=n//k)*(6*k**2*q+((k<<1)-1)*(q+1)*((q<<1)+1)) for k in range(1,s+1)))//6 # _Chai Wah Wu_, Oct 24 2023
%Y A350123 Cf. A064602, A143128, A222548, A350107, A350124, A350125, A350128.
%K A350123 nonn
%O A350123 1,2
%A A350123 _Seiichi Manyama_, Dec 15 2021