A327274 - OEIS

%I #28 Sep 15 2023 07:52:47

%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,

%T -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,

%U 4,0,-2,4,-2,0,-2,16,4,0,-2,-2,4,0,-2,2,-2,0,-2,-2,4,0,-2,-8

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

%C Dirichlet inverse of A048272.

%C Moebius transform of A067856.

%H Antti Karttunen, <a href="/A327274/b327274.txt">Table of n, a(n) for n = 1..65537</a>

%F a(1) = 1; a(n) = -Sum_{d|n, d<n} A048272(n/d) * a(d).

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

%F a(n) = 0 if n == 2 (mod 4). - _Bernard Schott_, Dec 07 2021

%F 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

%t 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}]

%t 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 *)

%o (PARI)

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

%o A327274(n) = sumdiv(n,d,moebius(n/d)*A067856(d));

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

%K sign,easy,mult

%O 1,3

%A _Ilya Gutkovskiy_, Oct 22 2019