Trigonometry: Double-Angle Identities

What is the sum of all positive integer values of $n$ that satisfy the equation $$\cos \Bigl( \frac{\pi}{n} \Bigr) \cos \Bigl( \frac{2\pi}{n} \Bigr) \cos \Bigl( \frac{4\pi}{n} \Bigr) \cos \Bigl( \frac{8\pi}{n} \Bigr) \cos \Bigl( \frac{16\pi}{n} \Bigr) = \frac{1}{32}?$$

(Hint: Does this remind you of the sine double angle formula?)

Source: Math Prize for Girls