For any positive integer
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
define
![g(x)](/media/m/4/b/0/4b0b04177e3643e2da7ebf1f112c0e69.png)
as greatest odd divisor of
![x,](/media/m/a/9/6/a96327826d8ff11119b89ba22bc1e70a.png)
and
Construct the sequence
![x_1 = 1, x_{n + 1} = f(x_n).](/media/m/d/4/2/d4220b22c4635547938fc9ab0864d97f.png)
Show that the number 1992 appears in this sequence, determine the least
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
such that
![x_n = 1992,](/media/m/f/8/e/f8eb7b3e1292596bc18f66f199be729a.png)
and determine whether
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
is unique.
%V0
For any positive integer $x$ define $g(x)$ as greatest odd divisor of $x,$ and $$f(x) =\begin{cases}\frac{x}{2}+\frac{x}{g(x)}&\text{if\ \(x\) is even},\\ 2^{\frac{x+1}{2}}&\text{if\ \(x\) is odd}.\end{cases}$$
Construct the sequence $x_1 = 1, x_{n + 1} = f(x_n).$ Show that the number 1992 appears in this sequence, determine the least $n$ such that $x_n = 1992,$ and determine whether $n$ is unique.