Question 2
Let $m$ be a fixed positive integer. Define an infinite sequence by a positive integer $a_1$ and, for every $n\ge1$,
$$a_{n+1}=\begin{cases}a_n^2+2^m,&a_n\lt 2^m,\\a_n/2,&a_n\ge2^m.\end{cases}$$
For each $m$, determine all $a_1$ for which every term is an integer.
$$a_{n+1}=\begin{cases}a_n^2+2^m,&a_n\lt 2^m,\\a_n/2,&a_n\ge2^m.\end{cases}$$
For each $m$, determine all $a_1$ for which every term is an integer.