On terms in a dynamical divisibility sequence having a fixed G.C.D with their indices

Abstract

Let $F$ and $G$ be integer polynomials where $F$ has degree at least $2$. Define the sequence $(a_n)$ by $a_n=F(a_{n-1})$ for all $n\ge 1$ and $a_0=0.$ Let ${\mathcal{B}}_{F,G,k}$ be the set of all positive integers $n$ such that $k\mid \gcd (G(n),a_n)$ and if $p\mid \gcd (G(n),a_n)$ for some $p$, then $p\mid k.$ Let ${\mathcal{A}}_{F,G,k}$ be the subset of ${\mathcal{B}}_{F,G,k}$ such that ${\mathcal{A}}_{F,G,k}=\{n\ge 1:\gcd (G(n),a_n)=k\}$. In this article, we prove that the asymptotic density of ${\mathcal{A}}_{F,G,k}$ and ${\mathcal{B}}_{F,G,k}$ exists for a class of $(F,G)$ and also compute the explicit density of ${\mathcal{A}}_{F,G,k}$ and ${\mathcal{B}}_{F,G,k}$ for $G(x)=x.$

arXiv commentImproved results and incorporated referee comments, 19 pages To appear in New York Journal of Mathematics