RU-ALLRUSSIAN 2021 Problem 92

Let $n$ be a natural number. An integer $a>2$ is called $n$ -decomposable, if $a^n-2^n$ is divisible by all the numbers of the form $a^d+2^d$ , where $d\neq n$ is a natural divisor of $n$ . Find all composite $n\in \mathbb{N}$ , for which there's an $n$ -decomposable number.