A374974 -id:A374974 - cp's OEIS frontend

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

Chai Wah Wu, Jul 25 2024

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

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

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

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

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

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

A376290 a(n) = Sum_{k=1..n-1} sigma_2(k) * sigma_3(n-k).

Original entry on oeis.org

0, 1, 14, 83, 324, 986, 2484, 5625, 11304, 21596, 37824, 64746, 103252, 163536, 244200, 364855, 517478, 741087, 1009244, 1394080, 1842690, 2470668, 3178188, 4171260, 5242610, 6735966, 8331338, 10511692, 12777898, 15922212, 19067506, 23429969, 27785000, 33707290

Offset: 1

Vaclav Kotesovec, Sep 19 2024

nonn

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).

Cf. A001157, A001158, A374974, A374963.

Table[Sum[DivisorSigma[2, k]*DivisorSigma[3, n-k], {k, n-1}], {n, 1, 50}]

a(n) = sum(k=1, n-1, sigma(k, 2) * sigma(n-k, 3)); \\ Michel Marcus, Sep 19 2024

Sum_{k=1..n} a(k) ~ Pi^4 * zeta(3) * n^7 / 37800.

cp's OEIS Frontend

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

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