HMMT 2004 U38

Let $S=\left\{p_{1} p_{2} \cdots p_{n} \mid p_{1}, p_{2}, \ldots, p_{n}\right.$ are distinct primes and $\left.p_{1}, \ldots, p_{n}<30\right\}$. Assume 1 is in $S$. Let $a_{1}$ be an element of $S$. We define, for all positive integers $n$ :

$$\begin{gathered} a_{n+1}=a_{n} /(n+1) \text { if } a_{n} \text { is divisible by } n+1 \\ a_{n+1}=(n+2) a_{n} \\ \text { if } a_{n} \text { is not divisible by } n+1 \end{gathered}$$

How many distinct possible values of $a_{1}$ are there such that $a_{j}=a_{1}$ for infinitely many $j$ 's?