A326814 Dirichlet g.f.: (1/zeta(s)) * Product_{p prime} (1 - 2 * p^(-s)).

1, -3, -3, 2, -3, 9, -3, 0, 2, 9, -3, -6, -3, 9, 9, 0, -3, -6, -3, -6, 9, 9, -3, 0, 2, 9, 0, -6, -3, -27, -3, 0, 9, 9, 9, 4, -3, 9, 9, 0, -3, -27, -3, -6, -6, 9, -3, 0, 2, -6, 9, -6, -3, 0, 9, 0, 9, 9, -3, 18, -3, 9, -6, 0, 9, -27, -3, -6, 9, -27, -3, 0, -3, 9, -6

Offset: 1

Author

Ilya Gutkovskiy, Oct 19 2019

Keywords

• sign
• mult
• easy

Comments

Moebius transform applied twice to A076479 (unitary Moebius function).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Ilya Gutkovskiy, Scatter plot of partial sums of A326814 up to n=10000.

Crossrefs

Cf. A001221, A007428, A008683, A046099 (positions of 0's), A076479, A182139 (Dirichlet inverse), A226177, A326415, A326815.

Programs

Mathematica

Table[Sum[MoebiusMu[n/d] MoebiusMu[d] 2^PrimeNu[d], {d, Divisors[n]}], {n, 1, 75}]
f[p_, e_] := Which[e == 1, -3, e == 2, 2, e > 2, 0]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 26 2020 *)

PARI

a(n) = sumdiv(n, d, moebius(n/d)*moebius(d)*2^omega(d)); \\ Michel Marcus, Oct 26 2020

PARI

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

Formula

a(n) = Sum_{d|n} mu(n/d) * mu(d) * 2^omega(d), where mu = A008683 and omega = A001221.

Multiplicative with a(p^e) = -3 if e = 1, 2 if e = 2, and 0 otherwise. - Amiram Eldar, Oct 26 2020