A363553 Möbius function of rank 5: a(n) = lambda(n) = A008836(n) if n is 5-free and 0 otherwise.

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

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

Cf. A008836, A363551, A363552.

Other generalizations of the Möbius function: A053864, A053865, A053981, A189021, A189022, A189023.

Sequence in context: A015280 A015156 A016303 * A053981 A189023 A016266

Adjacent sequences: A363550 A363551 A363552 * A363554 A363555 A363556

Keyword

sign,mult,easy

Author

Amiram Eldar, Jun 10 2023

Status

approved