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) and a₀=0. Let B_F,G,k be the set of all positive integers n such that k | gcd(G(n),a_n) and if p | gcd(G(n),a_n) for some p, then p | k. Let A_F,G,k be the subset of B_F,G,k such that A_F,G,k={n ≥ 1 : gcd(G(n),a_n)=k}. In this article, we prove that the asymptotic density of A_F,G,k and B_F,G,k exists for a class of (F,G) and also compute the explicit density of A_F,G,k and B_F,G,k for G(x)=x.

I would like to thank Emanuele Tron and Seoyoung Kim for looking at the article and providing valuable comments to improve its quality. I am thankful to Peter Mueller, David Speyer and Will Sawin for their answers on MathOverflow post [7] and Thomas Tucker for helpful discussions regarding Proposition 2.2 of the paper. I am grateful to Ayan Nath for his constant support and helpful advice. I am indebted to the anonymous referee for helpful comments.