International Mathematics Competition
for University Students
2017

IMC2017: Day 1, Problem 3

3. For any positive integer $m$, denote by $P\left(m\right)$ the product of positive divisors of $m$ (e.g. $P(6)=36$). For every positive integer $n$ define the sequence $$ a_1(n)=n, \qquad a_{k+1}(n)=P(a_k(n)) \quad (k=1,2,\ldots,2016). $$

Determine whether for every set $S\subseteq\{1,2,\ldots,2017\}$, there exists a positive integer $n$ such that the following condition is satisfied:

   For every $k$ with $1\le k\le 2017$, the number $a_k(n)$ is a perfect square if and only if $k\in S$.

Proposed by: Matko Ljulj, University of Zagreb

  Hint    Solution