A356125 a(n) = Sum_{k=1..n} k * sigma_2(k).

1, 11, 41, 125, 255, 555, 905, 1585, 2404, 3704, 5046, 7566, 9776, 13276, 17176, 22632, 27562, 35752, 42630, 53550, 64050, 77470, 89660, 110060, 126335, 148435, 170575, 199975, 224393, 263393, 293215, 336895, 377155, 426455, 471955, 540751, 591441, 660221, 726521

Offset: 1

Author

Seiichi Manyama, Jul 27 2022

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

Partial sums of A328259.

Column k=3 of A356124.

Cf. A319086, A356042.

Programs

Mathematica

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

PARI

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

PARI

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

PARI

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

Python

from math import isqrt
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

Formula

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

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

a(n) ~ zeta(3) * n^4 / 4. - Vaclav Kotesovec, Aug 02 2022