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A326814
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Dirichlet g.f.: (1/zeta(s)) * Product_{p prime} (1 - 2 * p^(-s)).
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2
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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, -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, 0, 9, 9, -3, 18, -3, 9, -6, 0, 9, -27, -3, -6, 9, -27, -3, 0, -3, 9, -6
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OFFSET
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1,2
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COMMENTS
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Moebius transform applied twice to A076479 (unitary Moebius function).
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LINKS
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Amiram Eldar, Table of n, a(n) for n = 1..10000
Ilya Gutkovskiy, Scatter plot of partial sums of A326814 up to n=10000.
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FORMULA
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a(n) = Sum_{d|n} mu(n/d) * mu(d) * 2^omega(d), where mu = A008683 and omega = A001221.
Multiplicative with a(p^e) = -3 if e = 1, 2 if e = 2, and 0 otherwise. - Amiram Eldar, Oct 26 2020
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MATHEMATICA
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Table[Sum[MoebiusMu[n/d] MoebiusMu[d] 2^PrimeNu[d], {d, Divisors[n]}], {n, 1, 75}]
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 *)
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PROG
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(PARI) a(n) = sumdiv(n, d, moebius(n/d)*moebius(d)*2^omega(d)); \\ Michel Marcus, Oct 26 2020
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - 2*X)*(1 - X))[n], ", ")) \\ Vaclav Kotesovec, Aug 22 2021
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CROSSREFS
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Cf. A001221, A007428, A008683, A046099 (positions of 0's), A076479, A182139 (Dirichlet inverse), A226177, A326415, A326815.
Sequence in context: A003560 A262212 A123676 * A122775 A086632 A038699
Adjacent sequences: A326811 A326812 A326813 * A326815 A326816 A326817
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KEYWORD
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sign,mult,easy
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AUTHOR
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Ilya Gutkovskiy, Oct 19 2019
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STATUS
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approved
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