A189022 Apostol's third-order Möbius function mu_3(n).

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

Offset

1

Comments

The function mu_k(n) is defined as 0, if a (k+1)st power of a prime divides n.

Otherwise it is (-1)^r where r is the number of distinct primes p that appear as p^k in the canonical factorization of n.

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Tom Apostol, Introduction to analytic number theory, (1976) Springer, page 50.

Tom Apostol, Mobius function of order k, Pac. J. Math. 32 (1) (1970) 21-27.

Antal Bege, A generalization of Apostol's Mobius functions of order k, arXiv:0907.5293 [math.NT], 2009.

Index entries for sequences computed from exponents in factorization of n.

Formula

mu_1(n) = mu(n) = A008683(n).

mu_k(n) = Sum_{d^k|n} mu_{k-1}(n/d^k)*mu_{k-1}(n/d), k>=2.

Sum_{k<=n} a(k) ~ c*n + O(n^(1/3) * log(n)), where c = Product_{p prime} (1 - 2/p^3 + 1/p^4) = 0.74469549790606742043... . - Amiram Eldar, Sep 18 2022

Maple

mu := proc(n, k) local d, a; if k = 1 then return numtheory[mobius](n) ; end if; a := 0 ; for d in numtheory[divisors](n) do if n mod (d^k) = 0 then a := a+procname(n/d^k, k-1)*procname(n/d, k-1) ; end if; end do: a ; end proc:
A189022 := proc(n) mu(n, 3) ; end proc:

Mathematica

Table[If[Max[FactorInteger[n][[All, 2]]] < 4, (-1)^Count[FactorInteger[n][[All, 2]], 3], 0], {n, 1, 100}] (* Geoffrey Critzer, Mar 03 2015 *)

Prog

(PARI) muk(n, k) = if (k==1, moebius(n), sumdiv(n, d, if (ispower(d, k), muk(n/d, k-1)*muk(n/sqrtnint(d, k), k-1), 0)));
vector(100, n, muk(n, 3)) \\ Michel Marcus, Mar 04 2015

Crossrefs

Cf. A008683, A046101 (n for which a(n)=0), A189021, A189023.

Sequence in context: A363552 A307430 A053865 * A370598 A166234 A074481

Adjacent sequences: A189019 A189020 A189021 * A189023 A189024 A189025

Keyword

sign,mult

Author

R. J. Mathar, Apr 15 2011

Status

approved