A299406 Dirichlet g.f.: Sum_{n>0} a(n)/n^s = (zeta(s)*zeta(6*s))/(zeta(2*s)*zeta(3*s)).

1, 1, 1, 0, 1, 1, 1, -1, 0, 1, 1, 0, 1, 1, 1, -1, 1, 0, 1, 0, 1, 1, 1, -1, 0, 1, -1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, -1, 1, 1, 1, 0, 0, 1, 1, -1, 0, 0, 1, 0, 1, -1, 1, -1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, -1, -1, 1, 1, 0, 1, 1, 1, -1, 1, 0, 1, 0, 1, 1

Offset

1,1

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Vaclav Kotesovec, Plot of Sum_{k=1..n} a(k) / (2*n*Pi^4/(315*Zeta(3))) for n = 1..1000000

Index entries for sequences computed from exponents in factorization of n

Formula

a(n) is multiplicative with a(p^e)=(-1)^(e mod 3 + e mod 6) if e mod 3 < 2, otherwise 0, p prime and e >= 0.

Dirichlet inverse b(n) is multiplicative with b(p^e) = (-1)^e if e < 3, otherwise 0, p prime and e >= 0.

Sum_{d|n} a(d)*A005361(n/d) = A000005(n).

Conjecture: a(n) = A008836(n) * A210826(n).

Sum_{k=1..n} a(k) ~ 2*n*Pi^4/(315*Zeta(3)). - Vaclav Kotesovec, Feb 08 2019

Mathematica

f[e_] := If[Mod[e, 3] < 2, (-1)^(Mod[e, 3] + Mod[e, 6]), 0];
a[n_] := a[n] = Times @@ (f /@ FactorInteger[n][[All, 2]]);
Array[a, 100] (* Jean-François Alcover, Feb 26 2018 *)

Prog

(PARI) A299406(n) = { my(es = factor(n)[, 2]); factorback(apply(e -> if(2==(e%3), 0, (-1)^((e%3)+(e%6))), es)); }; \\ Antti Karttunen, Jul 29 2018

Crossrefs

Cf. A000005, A005361, A008836, A210826.

Sequence in context: A307425 A210826 A307421 * A287769 A267866 A175087

Adjacent sequences: A299403 A299404 A299405 * A299407 A299408 A299409

Keyword

sign,mult

Author

Werner Schulte, Feb 20 2018

Extensions

Name clarified by Andrey Zabolotskiy, Dec 11 2023

Status

approved