A341831 Dirichlet g.f.: 1 / zeta(s)^5.

1, -5, -5, 10, -5, 25, -5, -10, 10, 25, -5, -50, -5, 25, 25, 5, -5, -50, -5, -50, 25, 25, -5, 50, 10, 25, -10, -50, -5, -125, -5, -1, 25, 25, 25, 100, -5, 25, 25, 50, -5, -125, -5, -50, -50, 25, -5, -25, 10, -50, 25, -50, -5, 50, 25, 50, 25, 25, -5, 250, -5, 25, -50, 0, 25

Offset: 1

Author

Ilya Gutkovskiy, Feb 21 2021

Keywords

• sign
• mult

Comments

Dirichlet inverse of A061200.

Links

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

Crossrefs

Row 5 of A346148.

Cf. A007427, A007428, A008683, A061200, A247343, A341832, A341833, A341834, A341835, A341836.

Programs

Mathematica

a[1] = 1; a[n_] := Times @@ ((-1)^#[[2]] Binomial[5, #[[2]]] &/@ FactorInteger[n]); Table[a[n], {n, 65}]

PARI

for(n=1, 100, print1(direuler(p=2, n, (1 - X)^5)[n], ", ")) \\ Vaclav Kotesovec, Feb 22 2021

Formula

Multiplicative with a(p^e) = (-1)^e * binomial(5, e).

a(1) = 1; a(n) = -Sum_{d|n, d < n} tau_5(n/d) * a(d).