Let $m$ be a positive integer number and $\left(a_n\right)$ be a sequence such that $a_1\in \mathbb{Z}^+$ and $$a_{n+1}=\left\{\begin{matrix} a_n^2 +2^m \quad &\text{if} \quad a_n<2^m,\\ \frac{a_n}{2} &\text{otherwise}. \end{matrix}\right.$$ For each $m\in \mathbb{Z}^+$, find all positive integer numbers $a_1$ such that $a_n$ is a integer number for all $n \in \mathbb{Z^+}$.
First I assume that $a_1 \geq 2^m$ and assume $a_1=2^kl$, where $l$ is an odd number. Then $l$ must be less than $2^m$ since if not, then $a_{k+1}=l$ and $a_{k+2}$ is not an integer number. And since the assumption $a_1 \geq 2^m$, there must exist an $h$ such that $a_{h+1}=2^{k-h}l<2^m$ and $a_h=2^{k-h+1}l >2^m$. Then $a_{h+2}=2^{k-h}l+2^m$ and $a_{h+3}=2^{k-h-1}l+2^{m-1}$. Since $2^{k-h}l<2^m$, we see that $a_{h+3}<2^{m-1}+2^{m-1}=2^m$ and therefore $a_{h+4}=2^{k-h-1}l+2^{m-1}+2^m$. Then $a_{h+5}=2^{k-h-2}l+2^{m-2}+2^{m-1}<2^{m-2}+2^{m-2}+2^{m-1}=2^m$. Going on the process till $k-h-1=0$ yields the next term will not be an integer. This implies that $l=0$.
I will show that if $a_1=2^m$, then $a_n$ is a integer number. Indeed, $a_2=2^{m-1}, a_3=2^{m-1}+2^m, a_4=2^{m-2}+2^{m-1}, a_5=2^{m-2}+2^{m-1}+2^m$. We obtain by induction that $a_{2k+1}=2^m+2^{m-1}+...+2^{m-k}$ and $a_{2k}=2^{m-1}+...+2^{m-k}$.
The case $a_1=2^k$ with $k>m$ will done when $a_{k-m+1}=2^m$
The case $a_1<2^m$ will turn into the first case that $a_2 \geq 2^m$ and $a_2=2^kl$ with an odd number $l$.
This completes the proof.
Am I on the right track? It may be quite "hand-crafted". Help me if there is a mistake or a better way. Thank you.
You should state what you are trying to prove. It appears the claim is that the only $a_1$ that works is $2^m$.
$l$ cannot be zero because then $a_1=0$, which is not allowed.
If you find any $a_1$ that is greater than or equal to $2^{m-1}$ you can multiply it by any power of $2$ and have a new $a_1$ that works. In particular, if $2^m$ works, so does any higher power of $2$.
$2^m$ does not work. $a_2=2^{m-1}$, then $a_3=3\cdot 2^{m-1}$ and every two iterations will remove a power of $2$ until you get to an odd number greater than $2^m$ and fail. For example, if $m=4$, the sequence is $16,8,24,12,28,14,30,15,31,31/2$