A363552 Möbius function of rank 4: a(n) = lambda(n) = A008836(n) if n is biquadratefree (A046100) and 0 otherwise.

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

Offset

1

Comments

Dirichlet inverse of A353519.

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

a(n) = A008836(n) * A307430(n).

abs(a(n)) = A307430(n).

Multiplicative with a(p^e) = (-1)^e if e <= 3, and 0 otherwise.

Dirichlet g.f.: Product_{p prime} (1 - 1/p^s + 1/p^(2*s) - 1/p^(3*s)) = zeta(2*s)/(zeta(s)*zeta(4*s)).

Mathematica

f[p_, e_] := If[e < 4, (-1)^e, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

(PARI) f(e) = if(e < 4, (-1)^e, 0);
a(n) = vecprod(apply(f, factor(n)[, 2]));

Crossrefs

Cf. A008683, A008836, A046100, A157295, A307430, A353519, A363551, A363553.

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

Sequence in context: A167851 A340369 A307445 * A307430 A053865 A189022

Adjacent sequences: A363549 A363550 A363551 * A363553 A363554 A363555

Keyword

sign,mult,easy

Author

Amiram Eldar, Jun 10 2023

Status

approved