Sums of Power Sums
Let $f_k(n)$ be the sum of the $k$th powers of the first $n$ positive integers.
For example, $f_2(10) = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 + 9^2 + 10^2 = 385$.
Let $S_k(n)$ be the sum of $f_k(i)$ for $1 \le i \le n$. For example, $S_4(100) = 35375333830$.
What is $\sum (S_{10000}(10^{12}) \bmod p)$ over all primes $p$ between $2 \cdot 10^9$ and $2 \cdot 10^9 + 2000$?