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%I #21 Aug 22 2021 05:55:34
%S 1,-3,-3,2,-3,9,-3,0,2,9,-3,-6,-3,9,9,0,-3,-6,-3,-6,9,9,-3,0,2,9,0,-6,
%T -3,-27,-3,0,9,9,9,4,-3,9,9,0,-3,-27,-3,-6,-6,9,-3,0,2,-6,9,-6,-3,0,9,
%U 0,9,9,-3,18,-3,9,-6,0,9,-27,-3,-6,9,-27,-3,0,-3,9,-6
%N Dirichlet g.f.: (1/zeta(s)) * Product_{p prime} (1 - 2 * p^(-s)).
%C Moebius transform applied twice to A076479 (unitary Moebius function).
%H Amiram Eldar, <a href="/A326814/b326814.txt">Table of n, a(n) for n = 1..10000</a>
%H Ilya Gutkovskiy, <a href="/A326814/a326814.jpg">Scatter plot of partial sums of A326814 up to n=10000</a>.
%F a(n) = Sum_{d|n} mu(n/d) * mu(d) * 2^omega(d), where mu = A008683 and omega = A001221.
%F Multiplicative with a(p^e) = -3 if e = 1, 2 if e = 2, and 0 otherwise. - _Amiram Eldar_, Oct 26 2020
%t Table[Sum[MoebiusMu[n/d] MoebiusMu[d] 2^PrimeNu[d], {d, Divisors[n]}], {n, 1, 75}]
%t f[p_, e_] := Which[e == 1, -3, e == 2, 2, e > 2, 0]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Oct 26 2020 *)
%o (PARI) a(n) = sumdiv(n, d, moebius(n/d)*moebius(d)*2^omega(d)); \\ _Michel Marcus_, Oct 26 2020
%o (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - 2*X)*(1 - X))[n], ", ")) \\ _Vaclav Kotesovec_, Aug 22 2021
%Y Cf. A001221, A007428, A008683, A046099 (positions of 0's), A076479, A182139 (Dirichlet inverse), A226177, A326415, A326815.
%K sign,mult,easy
%O 1,2
%A _Ilya Gutkovskiy_, Oct 19 2019
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