Question 7
Given a positive integer $m$, the sequence $\{a_n\}$ is defined as follows:
$$
a_1 = \frac{m}{2}, \quad a_{n+1} = a_n [a_n], \text{ if } n \ge 1.
$$
Determine all values of $m$ for which $a_{2007}$ is the first integer appearing in the sequence.
Note: For a real number $x$, $[x]$ denotes the smallest integer greater than or equal to $x$. For example, $[\pi] = 4$, $[2007] = 2007$.
$$
a_1 = \frac{m}{2}, \quad a_{n+1} = a_n [a_n], \text{ if } n \ge 1.
$$
Determine all values of $m$ for which $a_{2007}$ is the first integer appearing in the sequence.
Note: For a real number $x$, $[x]$ denotes the smallest integer greater than or equal to $x$. For example, $[\pi] = 4$, $[2007] = 2007$.