Problem #444

A big sum 2
Public 11/27/17 12xp Math 85.7%

Let \[S(m,n) = \sum\limits_{k=1}^m \sum\limits_{x=1}^n \big(x^k \cdot \lfloor \frac{{\pi x}}{k} \rfloor ^k \big)\] where $\lfloor x \rfloor$ is the largest integer less than or equal to $x$.

You are given $S(1, 10) = 1182$, $S(3, 100) = 16716537706541$ and $S(10, 10^4) \mod 1000000007 = 211872856$.

Find $S(20, 10^{20}) \mod 1000000007$.


Hint: solve Problem 399 - A Big Sum first and there are some clues in the secret forum.



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