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A189022
Apostol's third-order Möbius function mu_3(n).
6
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
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.
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
KEYWORD
sign,mult
AUTHOR
R. J. Mathar, Apr 15 2011
STATUS
approved