Let $g(x)$ be a polynomial with integral coefficients.


For $r\geq 1$, We define the sequence $a\_{g}$ for some polynomial $g(x)$ as follows:


$\clubsuit)a\_{g}(1)=g(x)$


$\clubsuit)a\_{g}(r)=g(a\_{g}(r-1))$ for $r\geq 2$


Now we are given a polynomial $f(x)$ of $\deg(f)\geq 2$ such that if $x=0$, $a\_{f}(r)\rightarrow\infty$ as $r$ grows large.


Then is it true that if $x=0$ then sum of reciprocal of primes $p$ such that $p|a\_{f}(p)$ converges?


**P.S.An analogous result has been proven for elliptic curves by Serre.**



Is it true that sum of reciprocal of primes $p$ such that $p|a_{f}(p)$ converges?
Have you seen [Silverman](https://arxiv.org/pdf/0707.1505.pdf) (Section 4)? The sum that he estimates there is the dynamical analogue of the relevant sums that appear when studying the analogue of $\gcd(n,f^n(0))$ (as in Kim for EDS, Sanna and myself for Lucas sequences... I am guessing this based on your post history). Otherwise I do not know about the exact sum you need, and I strongly suspect that such a bound is not in the literature - a search in the papers that cite Serre or Silverman turned out no results.


