## Sums of power sums

### Problem 487

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 ≤ `i` ≤ `n`. For example, S_{4}(100) = 35375333830.

What is ∑ (S_{10000}(10^{12}) mod p) over all primes `p` between 2 ⋅ 10^{9} and 2 ⋅ 10^{9} + 2000?