Question 40
The function $f$ is defined on the non-negative integers. $f\left(2^{n}-1\right)=0$ for $n=0,1,2, \ldots$. If $m$ is not of the form $2^{n}-1$, then $f(m)=f(m+1)+1$. Show that $f(n)+n=2^{k}-1$ for some $k$, and find $f\left(2^{1990}\right)$