Advanced Number Theory · Challenge problem

A function $f$ on the positive integers is defined by $f(1) = 1$, $f(2n) = f(n)$ and $f(2n+1) = f(n) + f(n+1)$ for every $n \ge 1$; so $f(2) = 1$, $f(3) = 2$, $f(4) = 1$, $f(5) = 3$, and so on. Find the remainder when $f(1) + f(2) + \cdots + f(1024)$ is divided by $1000$.