OIM 2007 Problem 1

Given an integer $m$ , define the sequence $\left\{a_{n}\right\}$ as follows: $$a_{1}=\frac{m}{2},\ a_{n+1}=a_{n}\left\lceil a_{n}\right\rceil,\textnormal{ if }n\geq 1$$ Find all values of $m$ for which $a_{2007}$ is the first integer appearing in the sequence.

Note: For a real number $x$ , $\left\lceil x\right\rceil$ is defined as the smallest integer greater or equal to $x$ . For example, $\left\lceil\pi\right\rceil=4$ , $\left\lceil 2007\right\rceil=2007$ .