MathNet 2010 — Part 03 Question 22

The sequence $(a_n)$ is defined by $a_1 = 1$ and $a_n = n \cdot (a_1 + \dots + a_{n-1})$ for all $n \gt 1$. Find all indices $n$ for which $a_n$ is divisible by $1 \cdot 2 \cdot \dots \cdot n$. (Grade 12.)