For each integer $n$ greater than 1, let $F(n)$ be the number of solutions of the equation $\sin x = \sin nx$ on the interval $[0, \pi]$.  What is $\sum_{n=2}^{2007} F(n)$?
Question:
For each integer $n$ greater than 1, let $F(n)$ be the number of solutions of the equation $\sin x = \sin nx$ on the interval $[0, \pi]$.  What is $\sum_{n=2}^{2007} F(n)$?
Answer:
Let's think step by step.

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2{,}016{,}532
For each integer $n$ greater than 1, let $F(n)$ be the number of solutions of the equation $\sin x = \sin nx$ on the interval $[0, \pi]$.  What is $\sum_{n=2}^{2007} F(n)$?
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