IL-GILLIS 1996 Problem 8

Consider the function $f : N \to N$ given by (i) $f(1) = 1$ , (ii) $f(2n) = f(n)$ for any $n \in N$ , (iii) $f(2n+1) = f(2n)+1$ for any $n \in N$ . (a) Find the maximum value of $f(n)$ for $1 \le n \le 1995$ ; (b) Find all values of $f$ on this interval.