Relatively Prime Sets, Divisor Sums, and Partial Sums

TL;DR

This paper studies the properties and sums of functions counting subsets of integers with gcd 1, introduces new interpretations and congruences for divisor sums related to these counts, and discusses open questions.

Contribution

It derives partial sums, provides combinatorial interpretations, and establishes congruence properties for functions related to gcd-1 subsets, introducing new insights into divisor sums and gcd-based combinatorics.

Findings

Partial sums of f(n), Φ(n), and D(n) are obtained.

A combinatorial interpretation of D(n) is provided.

A congruence property of D(n) is established.

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.