IMC 2019 Problem 7

Let $C=\{4,6,8,9,10,\ldots\}$ be the set of composite positive integers. For each $n\in C$ let $a_n$ be the smallest positive integer $k$ such that $k!$ is divisible by $n$ . Determine whether the following series converges: $$\sum_{n\in C}\left(\frac{a_n}{n}\right)^n.$$ Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan