OME 2001 Problem 6

Define the function $f: \mathbb{N} \rightarrow \mathbb{N}$ which satisfies, for any $s, n \in \mathbb{N}$ , the following conditions: $f(1) = f(2^s)$ and if $n < 2^s$ , then $f(2^s + n) = f(n) + 1.$ Calculate the maximum value of $f(n)$ when $n \leq 2001$ and find the smallest natural number $n$ such that $f(n) = 2001.$