On numbers $n$ with polynomial image coprime with the $n$th term of a linear recurrence

Abstract

Let $F$ be an integral linear recurrence, $G$ be an integer-valued polynomial splitting over the rationals, and $h$ be a positive integer. Also, let ${\mathcal{A}}_{F,G,h}$ be the set of all natural numbers $n$ such that $\gcd (F(n),G(n))=h$. We prove that ${\mathcal{A}}_{F,G,h}$ has a natural density. Moreover, assuming $F$ is non-degenerate and $G$ has no fixed divisors, we show that $\mathbf{d}({\mathcal{A}}_{F,G,1})=0$ if and only if ${\mathcal{A}}_{F,G,1}$ is finite.

References in corpus (3)

• The density of numbers $n$ having a prescribed G.C.D. with the $n$th Fibonacci number
• Distribution of integral values for the ratio of two linear recurrences
• The density of the terms in an elliptic divisibility sequence having a fixed G.C.D. with their index

Cited by in corpus (1)

• Greatest common divisors of shifted primes and Fibonacci numbers