Problem 487

Sums of power sums

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?

幂之和之和

记f_k(n)是前n个正整数的k次方的和。

例如，f₂(10) = 1² + 2² + 3² + 4² + 5² + 6² + 7² + 8² + 9² + 10² = 385。

记S_k(n)是所有满足1 ≤ i ≤ n的f_k(i)的和。例如，S₄(100) = 35375333830。

对于所有在2 ⋅ 10⁹与2 ⋅ 10⁹ + 2000之间的素数p，求∑ (S₁₀₀₀₀(10¹²) mod p)。