A327268 - OEIS

%I #29 Sep 15 2023 06:21:14

%S 1,1,-4,6,-6,-4,-8,24,3,-6,-12,-24,-14,-8,24,96,-18,3,-20,-36,32,-12,

%T -24,-96,5,-14,0,-48,-30,24,-32,384,48,-18,48,18,-38,-20,56,-144,-42,

%U 32,-44,-72,-18,-24,-48,-384,7,5,72,-84,-54,0,72,-192,80,-30,-60,144,-62,-32,-24,1536,84

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

%C Dirichlet inverse of A002129.

%H Georg Fischer, <a href="/A327268/b327268.txt">Table of n, a(n) for n = 1..1000</a>

%F a(1) = 1; a(n) = -Sum_{d|n, d<n} A002129(n/d) * a(d).

%F 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

%t 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}]

%t 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 *)

%Y Cf. A002129, A046692, A327274, A327278.

%K sign,easy,mult

%O 1,3

%A _Ilya Gutkovskiy_, Oct 22 2019