UA-UMO 1997 Problem 108

Let $d(n)$ denote the largest odd divisor of a positive integer $n$ . The function $f: \mathbb{N} \rightarrow \mathbb{N}$ is defined by $f(2n-1)=2^n$ and $f(2n)=n+\frac{2n}{d(n)}$ for all $n \in \mathbb{N}$ . Find all natural numbers $k$ such that: $f(f(...f(1)...))=1997.$ (where the paranthesis appear $k$ times)