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

TL;DR

This paper studies the distribution of indices in a polynomial-generated sequence where the gcd with a fixed polynomial value has a specific divisibility property, establishing the existence and explicit density of such indices.

Contribution

It proves the existence of asymptotic densities for sets of indices with fixed gcd properties in polynomial sequences and computes these densities explicitly for linear polynomials.

Findings

Asymptotic densities exist for the sets of interest.

Explicit formulas for densities are derived when G(x)=x.

Results apply to a class of polynomial pairs (F,G).

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 $\mathscr{B}_{F,\,G,\,k}$ 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 $\mathscr{A}_{F,\,G,\,k}$ be the subset of $\mathscr{B}_{F,\,G,\,k}$ such that $\mathscr{A}_{F,\,G,\,k}=\{n\ge 1 : \gcd(G(n),a_n)=k\}$. In this article, we prove that the asymptotic density of $\mathscr{A}_{F,\,G,\,k}$ and $\mathscr{B}_{F,\,G,\,k}$ exists for a class of $(F,G)$ and also compute the explicit density of $\mathscr{A}_{F,\,G,\,k}$ and $\mathscr{B}_{F,\,G,\,k}$ for $G(x)=x.$