Question 1
For each integer $a_0 \gt 1$, define the sequence $a_0$, $a_1$, $a_2$, …, by $$
a_{n+1} =
\begin{cases}
\sqrt{a_n} & \text{if $\sqrt{a_n}$ is an integer,} \\
a_n + 3 & \text{otherwise}
\end{cases}
$$ for each $n \ge 0$. Determine all values of $a_0$ for which there is a number $A$ such that $a_n = A$ for infinitely many values of $n$.
a_{n+1} =
\begin{cases}
\sqrt{a_n} & \text{if $\sqrt{a_n}$ is an integer,} \\
a_n + 3 & \text{otherwise}
\end{cases}
$$ for each $n \ge 0$. Determine all values of $a_0$ for which there is a number $A$ such that $a_n = A$ for infinitely many values of $n$.