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Problem #439

Enhanced perfect power
 Public ★(x8) 11/06/17 by liuguangxi 9xp Math 50.0%

A perfect power $n$ is called enhanced if there exists two integers $m>0$ and $k>2$ such that $n=m^k$. The enhanced perfect powers no more than $100$ are $1, 8, 16, 27, 32, 64, 81$.

Let $S(N)$ be the sum of all enhanced perfect powers no more than $N$. You are given $S(100) = 229$, $S(10000) = 99352$ and $S(10^8) \mod 1000000007 = 553225565$.

Find $S(10^{100}) \mod 1000000007$.

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