Coprime matchings of divisor sets

Abstract

For a positive integer $n$, let $D(n)$ be its set of positive divisors and put $N:=τ(n)=|D(n)|$. We characterize the arithmetic progressions $\mathcal P$ of length $N$ that admit a bijection with $D(n)$ pairing relatively prime integers. Such a coprime matching exists if and only if, for every prime $p\mid n$, at most $N/(ν_p(n)+1)$ terms of $\mathcal P$ are divisible by $p$, where $ν_p(n)$ is the exponent of $p$ in $n$. Call $n$ an $M$-number if $p^p\nmid n$ for every prime $p$. We prove that $D(n)$ can be matched with every primitive arithmetic progression of length $N$, meaning one whose first term and common difference are coprime, if and only if $n$ is an $M$-number. This proves a conjecture of McNew and Pomerance. We call $n$ \emph{matchable} if $D(n)$ admits a coprime matching with the primitive arithmetic progression $\{1,2,\ldots,N\}$. The general criterion implies that $n$ is matchable if and only if $\lfloor N/p\rfloor\le N/(ν_p(n)+1)$ for all $p\mid n$. We explicitly classify the matchable non-$M$-numbers, settle a conjectured divisor criterion, and prove that every divisor of a matchable integer is matchable. A Vandermonde determinant gives the required matchings under bounds on the prime exponents, while a partition into shorter progressions handles the remaining cases.