# On numbers $n$ relatively prime to the $n$th term of a linear recurrence

## TL;DR

This paper investigates the set of positive integers n for which a linear recurrence term u_n is coprime to n, establishing the existence of an asymptotic density and conditions for its positivity.

## Contribution

It proves the existence of an asymptotic density for the set of n where u_n and n are coprime, and characterizes when this density is positive.

## Key findings

- The set of such n has an asymptotic density.
- The density is positive unless (u_n / n) forms a linear recurrence.
- Provides conditions under which the density is zero or positive.

## Abstract

Let $(u_n)_{n \geq 0}$ be a nondegenerate linear recurrence of integers, and let $\mathcal{A}$ be the set of positive integers $n$ such that $u_n$ and $n$ are relatively prime. We prove that $\mathcal{A}$ has an asymptotic density, and that this density is positive unless $(u_n / n)_{n \geq 1}$ is a linear recurrence.