Let $a = \frac{\pi}{2008}$. Find the smallest positive integer $n$ such that
\[\sum_{k = 1}^n 2 \cos (k^2 a) \sin (ka)\]is an integer.
Question:
Let $a = \frac{\pi}{2008}$. Find the smallest positive integer $n$ such that
\[\sum_{k = 1}^n 2 \cos (k^2 a) \sin (ka)\]is an integer.
Answer:
Let's think step by step.

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Let $a = \frac{\pi}{2008}$. Find the smallest positive integer $n$ such that
\[\sum_{k = 1}^n 2 \cos (k^2 a) \sin (ka)\]is an integer.
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