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:

$$\sum_{n \in C} \left(\frac{a_n}{n}\right)^n. \tag{1}$$