IMC 2017 · Problem 3

For any positive integer $m$, denote by $P(m)$ 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 $k \in S$.