US-TST 2020 Problem 5

Find all integers $n \geq 2$ for which there exists an integer $m$ and a polynomial $P(x)$ with integer coefficients satisfying the following three conditions: - $m>1$ and $\operatorname{gcd}(m, n)=1$; - the numbers $P(0), P^{2}(0), \ldots, P^{m-1}(0)$ are not divisible by $n$; and - $P^{m}(0)$ is divisible by $n$. Here $P^{k}$ means $P$ applied $k$ times, so $P^{1}(0)=P(0), P^{2}(0)=P(P(0))$, etc.