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A363553 Möbius function of rank 5: a(n) = lambda(n) = A008836(n) if n is 5-free and 0 otherwise. 3
1, -1, -1, 1, -1, 1, -1, -1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, -1, 1, 1, -1, 1, 1, 1, -1, -1, -1, -1, -1, 0, 1, 1, 1, 1, -1, 1, 1, 1, -1, -1, -1, -1, -1, 1, -1, -1, 1, -1, 1, -1, -1, 1, 1, 1, 1, 1, -1, 1, -1, 1, -1, 0, 1, -1, -1, -1, 1, -1, -1, -1, -1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
1
COMMENTS
5-free numbers are numbers that are not divisible by a 5th power other than 1.
abs(a(n)) is the characteristic function of the 5-free numbers.
LINKS
Masato Kobayashi, Möbius functions of higher rank and Dirichlet series, arXiv:2108.01822 [math.NT], 2021.
FORMULA
Multiplicative with a(p^e) = (-1)^e if e <= 4, and 0 otherwise.
Dirichlet g.f.: Product_{p prime} (1 - 1/p^s + 1/p^(2*s) - 1/p^(3*s) + 1/p^(4*s)) = zeta(2*s)*zeta(5*s)/(zeta(s)*zeta(10*s)).
MATHEMATICA
f[p_, e_] := If[e < 5, (-1)^e, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) f(e) = if(e < 5, (-1)^e, 0);
a(n) = vecprod(apply(f, factor(n)[, 2]));
CROSSREFS
Other generalizations of the Möbius function: A053864, A053865, A053981, A189021, A189022, A189023.
Sequence in context: A015280 A015156 A016303 * A053981 A189023 A016266
KEYWORD
sign,mult,easy
AUTHOR
Amiram Eldar, Jun 10 2023
STATUS
approved

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Last modified April 15 14:37 EDT 2024. Contains 371689 sequences. (Running on oeis4.)