A053864 A second-order generalization of the Mobius function of n.

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

Offset

1,1

Comments

From R. J. Mathar, Apr 15 2011: (Start)

k-th order generalizations of the Mobius function are defined here (k=2), in A053865 (k=3) and A053981 (k=4) as follows:

The arithmetic function is 0 if the canonical prime power factorization of n contains an exponent > k. Otherwise, the function is 1 if any exponent is < k. In the remaining cases the function is (-1)^omega(n), where omega = A001221 gives the number of distinct primes dividing n.

The function is not multiplicative (example: a(3) = +1, a(4) = -1 and a(12) = +1). (End)

A212793(n) = abs(a(n)). - Reinhard Zumkeller, May 27 2012

Links

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

Maple

with(numtheory); k := 2; A053864 := proc(n, k) local t1, t2, i; t1 := 1; if n=1 then RETURN(t1); fi; t2 := factorset(n); for i in t2 do if n mod i^(k+1) = 0 then RETURN(0); fi; od; for i in t2 do if n mod i^k <> 0 then RETURN(1); else t1 := -t1; fi; od; t1; end;

Mathematica

a[n_] := Which[ee = FactorInteger[n][[All, 2]]; AnyTrue[ee, # > 2 &], 0, AnyTrue[ee, # < 2 &], 1, True, (-1)^Length[ee]]; Array[a, 100] (* Jean-François Alcover, Feb 09 2018 *)

Crossrefs

Cf. A008683, A053865, A053981.

Sequence in context: A115789 A359471 A363551 * A189021 A212793 A307420

Adjacent sequences: A053861 A053862 A053863 * A053865 A053866 A053867

Keyword

sign

Author

N. J. A. Sloane, Apr 08 2000

Status

approved