Question 14
Let $N^*$ be the set of all positive integers. Prove that there exists a unique function $f: N^* \to N^*$ satisfying $f(1) = f(2) = 1$ and $f(n) = f(f(n-1)) + f(n-f(n-1))$, $n = 3, 4, \dots$. For such $f$, find the value of $f(2^m)$ for integer $m \ge 2$.