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 $\left\{1,2,\dots ,n\right\}$ such that $\gcd \left(A\right)=1$ and the number of subsets $A$ of $\left\{1,2,\dots ,n\right\}$ 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.

submitted

References in corpus (1)

• Combinatorial Identities Involving the Mobius Function

Cited by in corpus (1)

• On some properties of the function of the number of relatively prime subsets of $\{1,2,...,n\}$