FI-MO 2019 Problem 4

Define a sequence $a_n = n^n + (n - 1)^{n+1}$ when $n$ is a positive integer. Define all those positive integer $m$ , for which this sequence of numbers is eventually periodic modulo $m$ , e.g. there are such positive integers $K$ and $s$ such that $a_k \equiv a_{k+s}$ ( $mod m$ ), where $k$ is an integer with $k \ge K$ .