Problem

Source: Turkey TST 2019 Day 1 P2

Tags: number theory, Sequence, prime numbers



$(a_{n})_{n=1}^{\infty}$ is an integer sequence, $a_{1}=1$, $a_{2}=2$ and for $n\geq{1}$, $a_{n+2}=a_{n+1}^{2}+(n+2)a_{n+1}-a_{n}^{2}-na_{n}$. $a)$ Prove that the set of primes that divides at least one term of the sequence can not be finite. $b)$ Find 3 different prime numbers that do not divide any terms of this sequence.