A note on generalized mobius (s-functions

In [1] the concept of a conjugate pair of sets of positive integers is introduced.  Briefly, if Z denotés the set of positive integers and P and Q denote non-empty subsets of Z such that: if  n₁  (pertenece a)  Z, n₂ (pertenece a) Z, (n₁,n₂) = 1, then

(1)  n = n₁n₂  (pertenece a)  P(resp. Q) <=> n₁  (pertenece a) P,n₂ (pertenece a) P (resp. Q),

and, if in addition, for each integer  n (pertenece a)  Z there is a unique factorization of the form

(2)  n = ab , a (pertenece a)  P, b (pertenece a)  Q,

we say that each of the sets P and Q is a direct factor set of Z, and that (P,Q) is a conjugate pair.

It is clear that  P (intersección) Q = {11}.  Among the generalized functions studied in [1] ,