CH-SMO 2022 Problem 5

For an integer $a \ge 2$ , denote by $\delta_(a)$ the second largest divisor of $a$ . Let $(a_n)_{n\ge 1}$ be a sequence of integers such that $a_1 \ge 2$ and $$a_{n+1} = a_n + \delta_(a_n)$$ for all $n \ge 1$ . Prove that there exists a positive integer $k$ such that $a_k$ is divisible by $3^{2022}$ .