# Primes And A Double Product

**Calculus**Level 5

\[\large f(n):=\prod_{a=1}^{p_n}\prod_{b=1}^{p_n}(1+\sqrt[p_n]{p_n}e^{2\pi i ab/p_n})\]

Let \(\{p_n\}_{n=1}^\infty=\{3,5,7,11,\ldots\}\) be the sequence of odd prime numbers in increasing order. Let us define the function above. Compute \[\sum_{n=1}^{2016}\left( \frac{\sqrt[p_n]{(1+p_n)f(n)}}{1+\sqrt[p_n]{p_n}} -p_n +n\right).\]