IMC 2014 · Problem 4

Let $n > 6$ be a perfect number, and let $n = p_1^{e_1} \cdots p_k^{e_k}$ be its prime factorisation with $1 < p_1 < \dots < p_k$. Prove that $e_1$ is an even number.

A number $n$ is perfect if $s(n) = 2n$, where $s(n)$ is the sum of the divisors of $n$.