PAMO 2018 Problem 3

For any positive integer $x$ , we set $$g(x) = \text{ largest odd divisor of } x,$$ $$f(x) = \begin{cases} \frac{x}{2} + \frac{x}{g(x)} & \text{ if } x \text{ is even;} 2^{\frac{x+1}{2}} & \text{ if } x \text{ is odd.} \end{cases}$$ Consider the sequence $(x_n)_{n \in \mathbb{N}}$ defined by $x_1 = 1$ , $x_{n + 1} = f(x_n)$ . Show that the integer $2018$ appears in this sequence, determine the least integer $n$ such that $x_n = 2018$ , and determine whether $n$ is unique or not.