r-Generalization of Phi Functions For The Subsets Of {m,m+1,...,n}

摘要

A nonempty finite set A of positive integers is r-relatively prime if greatest r~(th) power common divisor of elements of A is 1. In this case we write gcd_r (A) = 1.Let f~((r))(m,n) be the number of r-relatively prime subsets of {m, m + 1, ..., n} and the number of sets in f~((r))(m,n) of cardinality k is f_k~((r)) (m,n). The number of nonempty subsets which are r-relatively prime to n is Φ~((r))(m,n) and the number of sets in Φ~((r))(m,n) of cardinality k is Φ_k~((r)) (m,n). We obtained exact formulae and asymptotic estimates for these functions f~((r)) (m,n), f_k~((r)) (m, n), Φ~((r)) (m,n) and Φ_k~((r)) (m,n) in [4]. In this paper we find simple explicit formulae for these four functions which simplify the results in [4] and also find the asymptotic estimates for these functions.