A326938 Dirichlet g.f.: 1 / (zeta(s) * zeta(s-1) * (1 - 2^(-s))).

1, -2, -4, 0, -6, 8, -8, 0, 3, 12, -12, 0, -14, 16, 24, 0, -18, -6, -20, 0, 32, 24, -24, 0, 5, 28, 0, 0, -30, -48, -32, 0, 48, 36, 48, 0, -38, 40, 56, 0, -42, -64, -44, 0, -18, 48, -48, 0, 7, -10, 72, 0, -54, 0, 72, 0, 80, 60, -60, 0, -62, 64, -24, 0, 84, -96, -68, 0, 96, -96

Offset: 1

Author

Ilya Gutkovskiy, Oct 22 2019

Keywords

• sign
• mult

Comments

Dirichlet inverse of A002131.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002131, A008683, A046692, A327278.

Programs

Mathematica

a[1] = 1; a[n_] := -Sum[Total[Select[Divisors[n/d], OddQ[(n/d)/#] &]] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 70}]
Table[DivisorSum[n,  MoebiusMu[n/#] MoebiusMu[#] # &, OddQ[n/#] &], {n, 1, 70}]
f[2, e_] := -2*Boole[e == 1]; f[p_, e_] := Which[e == 1, -(p + 1), e == 2, p, e > 2, 0]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Nov 30 2020 *)

PARI

a(n)={sumdiv(n, d, if(n/d%2, moebius(n/d)*moebius(d)*d))} \\ Andrew Howroyd, Oct 25 2019

Formula

a(1) = 1; a(n) = -Sum_{d|n, dA002131(n/d) * a(d).

a(n) = Sum_{d|n, n/d odd} mu(n/d) * mu(d) * d.

Multiplicative with a(2^e) = -2 if e = 1 and 0 otherwise, and a(p^e) = -(p+1) for e = 1, p for e = 2 and 0 if e > 2, for odd primes p. - Amiram Eldar, Nov 30 2020