A321598 a(n) = Sum_{d|n} d*binomial(d+2,3).

1, 9, 31, 89, 176, 375, 589, 1049, 1516, 2384, 3147, 4823, 5916, 8437, 10406, 14105, 16474, 22380, 25271, 33264, 37810, 47683, 52901, 68183, 73301, 91100, 100174, 122197, 130356, 161750, 169137, 205593, 219162, 259242, 272714, 330524, 338144, 400719, 421686, 493424

Offset: 1

Author

Ilya Gutkovskiy, Nov 14 2018

Keywords

• nonn
• easy

Comments

Inverse Möbius transform of A002417.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• N. J. A. Sloane, Transforms.

Crossrefs

Cf. A000292, A000335, A001157, A001158, A001159, A002417, A013663, A059358, A278403.

Programs

Mathematica

Table[Sum[d Binomial[d + 2, 3], {d, Divisors[n]}], {n, 40}]
nmax = 40; Rest[CoefficientList[Series[Sum[x^k (1 + 3 x^k)/(1 - x^k)^5, {k, 1, nmax}], {x, 0, nmax}], x]]
Table[(2 DivisorSigma[2, n] + 3 DivisorSigma[3, n] + DivisorSigma[4, n])/6, {n, 40}]

PARI

a(n) = my(f = factor(n)); (sigma(f, 4) + 3*sigma(f, 3) + 2*sigma(f, 2)) / 6; \\ Amiram Eldar, Jan 02 2025

Formula

G.f.: Sum_{k>=1} x^k*(1 + 3*x^k)/(1 - x^k)^5.

G.f.: Sum_{k>=1} k*A000292(k)*x^k/(1 - x^k).

L.g.f.: -log(Product_{k>=1} (1 - x^k)^A000292(k)) = Sum_{n>=1} a(n)*x^n/n.

Dirichlet g.f.: (zeta(s-4) + 3*zeta(s-3) + 2*zeta(s-2))*zeta(s)/6.

a(n) = (2*sigma_2(n) + 3*sigma_3(n) + sigma_4(n))/6.

a(n) = Sum_{d|n} A002417(d).

Sum_{k=1..n} a(k) ~ zeta(5) * n^5 / 30. - Vaclav Kotesovec, Feb 02 2019