  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.
