Matchable numbers

TL;DR

This paper introduces the concept of matchable numbers, characterizes their distribution, proves all squarefree numbers are matchable, and discusses related open problems.

Contribution

It defines matchable numbers, computes their asymptotic density, and establishes that all squarefree numbers are matchable, advancing understanding of their properties.

Findings

The set of matchable numbers has a positive asymptotic density.

All squarefree numbers are matchable.

The paper presents open problems related to matchable numbers.

Abstract

We say a natural number $n$ is matchable if there is a bijection from the set of $\tau(n)$ divisors of $n$ to the set $\{1,2,\dots,\tau(n)\}$, where corresponding numbers are relatively prime. We show that the set of matchable numbers has an asymptotic density, which we compute, and we show that every squarefree number is matchable. We also present some related unsolved problems.