so I think we can let $a = m-1$ and write this as the problem:
Let $f(b) = b + k + a$, where $k$ and $a$ are non-negative integers satisfying $b = (2a+1) \cdot 2^k$.
Now consider the iterated application of the function $f^{n+1}(b) = f(f^n(b))$, $f^0(b) = b$. Does there exist a non-negative integer $n$ such that $f^n(5)$ equals a power of 2 or 3 times a power of 2?