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Problem 372

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Pencils of rays

Let R(M, N) be the number of lattice points (x, y) which satisfy M<x≤N, M<y≤N and $\lfloor \frac{y^2}{x^2}\rfloor$ is odd.
We can verify that R(0, 100) = 3019 and R(100, 10000) = 29750422.
Find R(2·10⁶, 10⁹).

Note : $\lfloor x\rfloor$ represents the floor function.

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光锥

记R(M, N)为满足M<x≤N、M<y≤N且$\lfloor \frac{y^2}{x^2}\rfloor$为奇数的格点(x, y)的数目。
我们可以验证R(0, 100) = 3019以及R(100, 10000) = 29750422。
求R(2·10⁶, 10⁹)。

注：$\lfloor x\rfloor$表示下取整函数。

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