How often is $x\mapsto x^3$ one-to-one in $\mathbb{Z}/n\mathbb{Z}$?

Abstract

We characterize the integers n such that $x\mapsto x^3$ describes a bijection from the set $\mathbb{Z}/n\mathbb{Z}$ to itself and we determine the frequency of these integers. Precisely, denoting by $W$ the set of these integers, we prove that an integer belongs to $W$ if and only if it is square-free with no prime factor that is congruent to 1 modulo 3, and that there exists $C>0$ such that $$|W\cap\{1,\dots,n\}|\sim C\frac{n}{\sqrt{\log n}}\ .$$ These facts (or equivalent facts) are stated without proof on the OEIS website. We give the explicit value of $C$, which did not seem to be known. Analogous results are also proved for families of integers for which congruence classes for prime factors are imposed. The proofs are based on a Tauberian Theorem by Delange.