A327274 Dirichlet g.f.: 1 / (zeta(s)^2 * (1 - 2^(1 - s))).

1, 0, -2, 1, -2, 0, -2, 2, 1, 0, -2, -2, -2, 0, 4, 4, -2, 0, -2, -2, 4, 0, -2, -4, 1, 0, 0, -2, -2, 0, -2, 8, 4, 0, 4, 1, -2, 0, 4, -4, -2, 0, -2, -2, -2, 0, -2, -8, 1, 0, 4, -2, -2, 0, 4, -4, 4, 0, -2, 4, -2, 0, -2, 16, 4, 0, -2, -2, 4, 0, -2, 2, -2, 0, -2, -2, 4, 0, -2, -8

Offset: 1

Author

Ilya Gutkovskiy, Oct 22 2019

Keywords

• sign
• easy
• mult

Comments

Dirichlet inverse of A048272.

Moebius transform of A067856.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A007427, A008683, A048272, A062503 (positions of 1's), A067856, A327268.

Programs

Mathematica

a[1] = 1; a[n_] := Sum[Sum[(-1)^j, {j, Divisors[n/d]}] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 80}]
f[p_, e_] := Switch[e, 1, -2, 2, 1, , 0]; f[2, e] := 2^(e-2); f[2, 1] = 0; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2023 *)

PARI

A067856(n) = { my(k); if(n<1, 0, k=valuation(n, 2); moebius(n/2^k)*2^max(0, k-1)); }; \\ From A067856
A327274(n) = sumdiv(n,d,moebius(n/d)*A067856(d));

Formula

a(1) = 1; a(n) = -Sum_{d|n, dA048272(n/d) * a(d).

a(n) = Sum_{d|n} mu(n/d) * A067856(d).

a(n) = 0 if n == 2 (mod 4). - Bernard Schott, Dec 07 2021

Multiplicative with a(2) = 0, a(2^e) = 2^(e-2) for e >= 2, and for an odd prime p, a(p) = -2, a(p^2) = 1, and a(p^e) = 0 for e >= 3. - Amiram Eldar, Sep 15 2023