A356126 a(n) = Sum_{k=1..n} k * sigma_3(k).

1, 19, 103, 395, 1025, 2537, 4945, 9625, 16438, 27778, 42430, 66958, 95532, 138876, 191796, 266692, 350230, 472864, 603204, 787164, 989436, 1253172, 1533036, 1926156, 2319931, 2834263, 3386143, 4089279, 4796589, 5749149, 6672701, 7871069, 9101837, 10605521

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 A281372.

Column k=4 of A356124.

Cf. A356043.

Programs

Mathematica

a[n_] := Sum[k * DivisorSigma[3, k], {k, 1, n}]; Array[a, 34] (* Amiram Eldar, Jul 28 2022 *)

PARI

a(n) = sum(k=1, n, k*sigma(k, 3));

PARI

a(n) = sum(k=1, n, k^4*binomial(n\k+1, 2));

PARI

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, k^4*x^k/(1-x^k)^2)/(1-x))

Python

from math import isqrt
def A356126(n): return ((-(s:=isqrt(n))**2*(s+1)**2*((s<<1)+1)*(s*(3*(s+1))-1)>>1)+sum(k*(q:=n//k)*(q+1)*(15*k**3+((q<<1)+1)*(q*(3*(q+1))-1)) for k in range(1,s+1)))//30 # Chai Wah Wu, Oct 24 2023

Formula

a(n) = Sum_{k=1..n} k^4 * binomial(floor(n/k)+1,2).

G.f.: (1/(1-x)) * Sum_{k>=1} k^4 * x^k/(1 - x^k)^2.