  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
$G(x)=x.$
