OIM 1990 Problem 1

Let $f$ be a function defined for the non-negative integers, such that:

a) $f(n)=0$ if $n=2^{j}-1$ for some $j \geq 0$ . b) $f(n+1)=f(n)-1$ otherwise.

i) Show that for every $n \geq 0$ there exists $k \geq 0$ such that $f(n)+n=2^{k}-1$ . ii) Find $f(2^{1990})$ .