A perfect number is a natural number $n$ that is the sum of its proper divisors. Otherwise said, they are the natural numbers $n$ such that $\sum_{d|n}d=2n.$ For instance, $6$ and $28$ are perfect.
Let $n$ be a perfect number, and for each integer $m\geq1$ define $a_{m}=-2+\sum_{d|m}d^m.$
I am looking for hints to prove the following claim:
There are at most finitely many primes that do not divide any of the $a_{m}.$