MathNet 2008-part-05 — 164 problems (EN–VI) Question 70

Let $f(n)$ be the number of times you have to hit the $\sqrt{\ }$ key on a calculator to get a number less than $2$ starting from $n$. For instance, $f(2)=1$, $f(5)=2$. For how many $1 \lt m \lt 2008$ is $f(m)$ odd?