USAMO 2006 — Days 1 and 2 Question 4

Find all positive integers $n$ for which there exist an integer $k \ge 2$ and positive rational numbers $a_1$, …, $a_k$ satisfying $a_1 + a_2 + \dots + a_k = a_1 a_2 \dots a_k = n$.

Proposed by Ricky Liu (credited in the supplied source).