Putnam 2013 A2

Let $S$ be the set of all positive integers that are not perfect squares. For $n \in S$, consider choices of integers $a_1, a_2, \ldots, a_r$ such that $n < a_1 < a_2 < \cdots < a_r$ and $n \cdot a_1 \cdot a_2 \cdots a_r$ is a perfect square, and let $f(n)$ be the minimum of $a_r$ over all such choices. Show that the function $f$ from $S$ to the integers is one-to-one.