A327268 Dirichlet g.f.: 1 / (zeta(s) * zeta(s-1) * (1 - 2^(2 - s))).

1, 1, -4, 6, -6, -4, -8, 24, 3, -6, -12, -24, -14, -8, 24, 96, -18, 3, -20, -36, 32, -12, -24, -96, 5, -14, 0, -48, -30, 24, -32, 384, 48, -18, 48, 18, -38, -20, 56, -144, -42, 32, -44, -72, -18, -24, -48, -384, 7, 5, 72, -84, -54, 0, 72, -192, 80, -30, -60, 144, -62, -32, -24, 1536, 84

Offset: 1

Author

Ilya Gutkovskiy, Oct 22 2019

Keywords

• sign
• easy
• mult

Comments

Dirichlet inverse of A002129.

Links

• Georg Fischer, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A002129, A046692, A327274, A327278.

Programs

Mathematica

a[1] = 1; a[n_] := Sum[Sum[(-1)^j j, {j, Divisors[n/d]}] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 65}]
f[p_, e_] := Switch[e, 1, -(p+1), 2, p, , 0]; f[2, e] := 3*2^(2*e-3); f[2, 1] = 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2023 *)

Formula

a(1) = 1; a(n) = -Sum_{d|n, dA002129(n/d) * a(d).

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