International Mathematics Competition
for University Students
2019

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

Hint: Find an upper bound on \(\displaystyle \frac{a_n}{n}\).

  Solution