Let $n$ be the number of polynomial functions from the integers modulo $2010$ to the integers modulo $2010$.  $n$ can be written as $n = p_1 p_2 \cdots p_k$, where the $p_i$s are (not necessarily distinct) primes.  Find $p_1 + p_2 + \cdots + p_n$.

Let's think step by step and output the final answer within \boxed{}.
user