Let f_k(n) be the sum of the k^th powers of the first n positive integers.

For example, f₂(10) = 1² + 2² + 3² + 4² + 5² + 6² + 7² + 8² + 9² + 10² = 385.

Let S_k(n) be the sum of f_k(i) for 1 ≤ i ≤ n. For example, S₄(100) = 35375333830.

What is ∑ (S₁₀₀₀₀(10¹²) mod p) over all primes p between 2 ⋅ 10⁹ and 2 ⋅ 10⁹ + 2000?