A189023 Apostol's fourth-order Mobius (Moebius) function mu_4(n).

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, 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

Offset

1

Links

R. J. Mathar, Table of n, a(n) for n = 1..1000

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^(1/4) * log(n)), where c = Product_{p prime} (1 - 2/p^4 + 1/p^5) = 0.88413948662046403031... . - 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:
A189023 := proc(n) mu(n, 4) ; end proc:

Mathematica

mu[n_, 1] := MoebiusMu[n]; mu[n_, k_] := mu[n, k] = Sum[Boole[Mod[n, d^k] == 0]*mu[n/d^k, k - 1]*mu[n/d, k - 1], {d, Divisors[n]}];
Table[mu[n, 4], {n, 1, 100}] (* Jean-François Alcover, Nov 24 2017 *)

Crossrefs

Cf. A008683, A189021, A189022.

Sequence in context: A016303 A363553 A053981 * A016266 A016192 A015970

Adjacent sequences: A189020 A189021 A189022 * A189024 A189025 A189026

Keyword

sign,mult

Author

R. J. Mathar, Apr 15 2011

Status

approved