This question is inspired by a [riddle][1] in math.stackexchange.

Let $P(x)$ be a polynomial, and $O = \{P^{(n)}(0) : n \geq 0\}$ be its orbit under zero (viewed as a set). Suppose that $O$ contains infinitely many integers. Is it true that for some $n$, $P^{(n)}$ is a polynomial with integral coefficients?

We can ask the same question replacing integers with rationals.


  [1]: http://math.stackexchange.com/questions/8101/iterated-polynomial-problem