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

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$\mathscr{A}{F,,G,,k}$bethesubsetof$\mathscr{B}{F,,G,,k}$suchthat$\mathscr{A}_{F,,G,,k}={n\ge 1 : \gcd(G(n),a_n)=k}$. 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 $G(x)=x\mathrm{.}$