Problem 262

The function $f$ is defined on the set of positive integers by $f(1) = 1$, $f(2n) = 2f(n)$, and $nf(2n + 1) = (2n + 1)(f(n) + n)$ for all $n \ge 1$.

i) Prove that $f(n)$ is always an integer.

ii) For how many positive integers less than $2007$ is $f(n) = 2n$?