SILK-ROAD 2018 Problem 4

Does there exist a sequence of positive integers $a_1,a_2,...$ such that every positive integer occurs exactly once and that the number $\tau (na_{n+1}^n+(n+1)a_n^{n+1})$ is divisible by $n$ for all positive integer.

Here $\tau (n)$ denotes the number of positive divisor of $n$ .