International Mathematics Competition
for University Students
2019

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IMC 2019
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IMC2019: Day 2, Problem 7

Problem 7.

Let \(\displaystyle C=\{4,6,8,9,10,\ldots\}\) be the set of composite positive integers. For each \(\displaystyle n\in C\) let \(\displaystyle a_n\) be the smallest positive integer \(\displaystyle k\) such that \(\displaystyle k!\) is divisible by \(\displaystyle n\). Determine whether the following series converges:

\(\displaystyle \sum_{n\in C}\Bigl(\frac{a_n}{n}\Bigr)^n . \)\(\displaystyle (1) \)

Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan

        

 

IMC
2019

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