Školjka

Let $a_1, a_2, \ldots$ be a sequence of positive integers such that $$a_1 = 1 \text{ and } a_{k+1} = a_k^3+1, \forall k \in\mathbb{N}$$ Prove that for every prime number $p$ of the form $3t+2$, where $t$ is a non-negative integer, there exists a positive integer $n$ such that $a_n$ is divisible by $p$.