Relatively Prime Sets, Divisor Sums, and Partial Sums

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