Let $f(n)$ be the number of divisors of $2n^2$ that are no greater than n. For example, $f(15)=8$ because there are 8 such divisors: 1,2,3,5,6,9,10,15. Note that 18 is also a divisor of $2\times 15^2$ but it is not counted because it is greater than 15.

Let $\displaystyle F(N) = \sum_{n=1}^N f(n)$. You are given $F(15)=63$, and $F(1000)=15066$.

Find $F(2^N)$.