IMO 1998 Problem 3

A function $f$ is defined on the positive integers by: $f(1) = 1$ ; $f(3) = 3$ ; $f(2n) = f(n)$ , $f(4n + 1) = 2f(2n + 1) - f(n)$ , and $f(4n + 3) = 3f(2n + 1) - 2f(n)$ for all positive integers $n$ . Determine the number of positive integers $n \le 1988$ for which $f(n) = n$ .

Hint: try converting the domain and range into binary.