IKMC 2013 - Grades 11-12 - Paper 02 - Self-study Question 25

For natural numbers, define $f(n)=n/2$ if $n$ is even and $f(n)=(n-1)/2$ if odd. For positive $k$, $f^k$ means applying $f$ exactly $k$ times. How many solutions does $f^{2013}(n)=1$ have?