SILK-ROAD 2020 Problem 1

Given a strictly increasing infinite sequence of natural numbers $a_1,$ $a_2,$ $a_3,$ $\ldots$ . It is known that $a_n \leq n + 2020$ and the number $n ^ 3 a_n - 1$ is divisible by $a_ {n + 1}$ for all natural numbers $n$ . Prove that $a_n = n$ for all natural numbers $n$ .