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Problem #438

Special squarefree sum
 Public ★(x5) 11/13/17 by liuguangxi 12xp Math 83.3%

A squarefree number is called special if it can be expressed as the sum of two positive squares. For example, $65( = 1^2+8^2)$ is a special squarefree number while $15$ is not.

Let $S(N)$ be the sum of all special squarefree numbers from $1$ to $N$. You are given that $S(10) = 17$, $S(1000) = 81665$, $S(10^6) = 61081126084$.

Find $S(10^{12})$.

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