Relatively Prime Sets, Divisor Sums, and Partial Sums

Abstract

For a nonempty finite set A of positive integers, let gcd (A) denote the greatest common divisor of the elements of A. Let f (n) and Φ (n) denote, respectively, the number of subsets A of {1, 2, . . . , n} such that gcd (A) = 1 and the number of subsets A of {1, 2, . . . , n} such that gcd (A ∪ {n}) = 1. Let D (n) be the divisor sum of f (n). In this article, we obtain partial sums of f (n), Φ (n) and D (n). We also obtain a combinatorial interpretation and a congruence property of D (n). We give open questions concerning Φ (n) and D (n) at the end of this article.