Question 126
Let $S=\{p_{1} p_{2} \cdots p_{n} \mid p_{1}, p_{2}, \ldots, p_{n}$ are distinct primes and $p_{1}, \ldots, p_{n}\lt 30\}$. Assume $1$ is in $S$. Let $a_{1}$ be an element of $S$. We define, for all positive integers $n$:
$$
\begin{gathered}
a_{n+1}=a_{n} /(n+1) \quad \text{if} a_{n} \text{is divisible by} n+1 \\
a_{n+1}=(n+2) a_{n} \\
\text{if} a_{n} \text{is not divisible by} n+1
\end{gathered}
$$
How many distinct possible values of $a_{1}$ are there such that $a_{j}=a_{1}$ for infinitely many $j$'s?
$$
\begin{gathered}
a_{n+1}=a_{n} /(n+1) \quad \text{if} a_{n} \text{is divisible by} n+1 \\
a_{n+1}=(n+2) a_{n} \\
\text{if} a_{n} \text{is not divisible by} n+1
\end{gathered}
$$
How many distinct possible values of $a_{1}$ are there such that $a_{j}=a_{1}$ for infinitely many $j$'s?