A374963 a(n) = Sum_{k=1..n-1} sigma(k)*sigma_3(n-k).

0, 1, 12, 59, 200, 526, 1184, 2399, 4368, 7656, 12316, 19586, 29008, 43244, 60272, 85543, 114000, 156163, 200652, 266504, 333968, 432570, 528704, 673706, 806200, 1008644, 1192584, 1467684, 1707328, 2084676, 2390848, 2882487, 3286168, 3913722, 4409584, 5237489

Offset: 1

Author

Chai Wah Wu, Jul 25 2024

Keywords

• nonn
• easy

Comments

Convolution of sigma with sigma_3.

In general, if k>=1, m>=1 and a(n) = Sum_{j=1..n-1} sigma_k(j) * sigma_m(n-j), then Sum_{j=1..n} a(j) ~ Gamma(k+1) * Gamma(m+1) * zeta(k+1) * zeta(m+1) * n^(k+m+2) / Gamma(k+m+3). - Vaclav Kotesovec, Sep 19 2024

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000203, A000385, A001158, A374974, A376290.

Programs

Mathematica

Table[Sum[DivisorSigma[1,k] *DivisorSigma[3,n-k],{k,n-1}],{n,36}] (* James C. McMahon, Aug 11 2024 *)

Python

from sympy import divisor_sigma
def A374963(n): return sum(divisor_sigma(i)*divisor_sigma(n-i,3) for i in range(1,n))

Formula

Sum_{k=1..n} a(k) ~ Pi^6 * n^6 / 64800. - Vaclav Kotesovec, Sep 19 2024