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A series over composite numbers

Source: IMC 2019 Day 2 P7

July 31, 2019
IMCConvergencenumber theorycollege contests

Problem Statement

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