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

Abstract

Let $(u_n{)}_{n\ge 0}$ 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 $(u_n/n{)}_{n\ge 1}$ is a linear recurrence.

References in corpus (2)

• On the greatest common divisor of $n$ and the $n$th Fibonacci number
• Distribution of integral values for the ratio of two linear recurrences

Cited by in corpus (3)

• On the greatest common divisor of $n$ and the $n$th Fibonacci number
• The density of numbers $n$ having a prescribed G.C.D. with the $n$th Fibonacci number
• Greatest common divisors of shifted primes and Fibonacci numbers