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A non-empty subset A of positive integers {1,2,..., n} is said to be relatively prime if ( ) gcd 1 A = . Let r be a positive integer 1. ≥ A nonempty subset { } 1,2,..., A n ⊆ is r-relatively prime if greatest th r power common divisor of elements of A is 1. In this case we write ( ) gcd 1. r A = Note that ( ) gcd 1 A = implies ( ) gcd 1 r A = but the converse need not be true. Let ( ) ( ) r f n denotes the number of rrelatively prime subsets of { } 1, 2, ..., n and ( ) ( ) r k f n denotes the number of r-relatively prime subsets of { } 1, 2, ..., n of Cardinality k. ( ) n Φ denotes the number of non empty subsets A of { } 1, 2, ..., n such that ( ) gcd A is relatively prime to n. ( ) k n Φ denotes the number of non-empty subsets A of Cardinality k of {1,2,...,n} such that ( ) gcd A is relatively prime to n. We define ( ) ( ) r n Φ to be the number of non-empty subsets A of { } 1, 2, ..., n such that greatest th r power common divisor of elements of A and n is 1. ( ) ( ) r k n Φ is defined as the number of subsets A of { } 1, 2, ..., n such that ( ) Card A k = and ( ) gcdr A is r-relatively to n. ( ) ( ) r n Φ and ( ) ( ) r k n Φ are r-generalizations of ( ) n Φ and ( ) k n Φ defined by Nathanson [2]. Exact formulas and asymp totic estimates are obtained for these functions. These results are ex t nsions of results of Nathanson [2]. Some of our proofs use the r-Generalization of Mobi us inversion formula.

r-Relatively Prime Sets and r-Generalization of Phi Functions for Subsets of {1,2,…,n}