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A307423 Dirichlet g.f.: zeta(2*s) / zeta(3*s). 4
1, 0, 0, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1 (list; graph; refs; listen; history; text; internal format)
OFFSET
1
COMMENTS
Dirichlet convolution of A212793 and A008836.
LINKS
Eric Weisstein's World of Mathematics, Dirichlet Generating Function.
Wikipedia, Dirichlet series.
FORMULA
Multiplicative with a(p) = 0, and a(p^e) = (-1)^e for e >= 2. - Amiram Eldar, Dec 25 2022
MATHEMATICA
Table[DivisorSum[n, Boole[Max[FactorInteger[#][[All, 2]]] < 3] * LiouvilleLambda[n/#]&], {n, 1, 100}]
f[p_, e_] := (-1)^e; f[p_, 1] := 0; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 25 2022 *)
PROG
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1-X^3)/(1-X^2))[n], ", ")) \\ Vaclav Kotesovec, Jun 14 2020
(PARI)
A212793(n) = factorback(apply(e->(e<=2), factor(n)[, 2]));
A307423(n) = sumdiv(n, d, ((-1)^bigomega(d))*A212793(n/d)); \\ Antti Karttunen, Jul 14 2022
CROSSREFS
Cf. A112526 (absolute values).
Sequence in context: A354868 A075802 A112526 * A355684 A355683 A120523
KEYWORD
sign,mult
AUTHOR
Vaclav Kotesovec, Apr 08 2019
EXTENSIONS
Data section extended up to a(108) by Antti Karttunen, Jul 14 2022
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)