%I #26 Oct 22 2023 00:47:56

%S 1,11,41,125,255,555,905,1585,2404,3704,5046,7566,9776,13276,17176,

%T 22632,27562,35752,42630,53550,64050,77470,89660,110060,126335,148435,

%U 170575,199975,224393,263393,293215,336895,377155,426455,471955,540751,591441,660221,726521

%N a(n) = Sum_{k=1..n} k * sigma_2(k).

%H Seiichi Manyama, <a href="/A356125/b356125.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{k=1..n} k^3 * binomial(floor(n/k)+1,2).

%F G.f.: (1/(1-x)) * Sum_{k>=1} k^3 * x^k/(1 - x^k)^2.

%F a(n) ~ zeta(3) * n^4 / 4. - _Vaclav Kotesovec_, Aug 02 2022

%t a[n_] := Sum[k * DivisorSigma[2, k], {k, 1, n}]; Array[a, 39] (* _Amiram Eldar_, Jul 28 2022 *)

%o (PARI) a(n) = sum(k=1, n, k*sigma(k, 2));

%o (PARI) a(n) = sum(k=1, n, k^3*binomial(n\k+1, 2));

%o (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, k^3*x^k/(1-x^k)^2)/(1-x))

%o (Python)

%o from math import isqrt

%o def A356125(n): return (-((s:=isqrt(n))*(s+1))**3>>1) + sum(k*(q:=n//k)*(q+1)*(2*k**2+q*(q+1)) for k in range(1,s+1))>>2 # _Chai Wah Wu_, Oct 21 2023

%Y Partial sums of A328259.

%Y Column k=3 of A356124.

%Y Cf. A319086, A356042.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Jul 27 2022