Question 104
For every integer $k \geqslant 1$, let $p(k)$ denote the smallest prime number not dividing $k$. Let $\left(a_{n}\right)_{n \in \mathbb{N}}$ be a sequence such that $a_{0} \in \mathbb{N}^{*}$ and, for every $n \geqslant 0$, $a_{n+1}$ is the smallest positive integer different from $a_{0}, \ldots, a_{n}$ such that $a_{n}^{a_{n+1}}-1$ is divisible by $p\left(a_{n}\right)$. Prove that every positive integer appears exactly once in the sequence $\left(a_{n}\right)_{n \in \mathbb{N}}$.