P5174 Dot Circles

The senior high school entrance exam is here. Pg stared at the last (hardest) problem on the math paper and fell into deep thought…… Solution: https://blog.csdn.net/kkkksc03/article/details/85008125

As everyone knows, every lattice point (integer-coordinate point) on the plane is passed through by **exactly one** circle centered at the origin. We define the weight of this lattice point as the **square of the radius** of the circle that passes through it. Now there is a circle centered at the origin with radius $\sqrt{R}$. You need to find the **sum** of the weights of all lattice points contained in this circle. Since the answer may be very large, output it modulo $10^9+7$.

One number $R$.

One line with only one number.

Explanation for Sample 1: There are $9$ lattice points in total: $(0,0),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1),(-1,0),(-1,1)$. Their weights are $\ 0,1,2,1,2,1,2,1,2$, and the total sum is 12. -------- Constraints: For $50\%$ of the testdata, $R \le 10^{6}.$ For $100\%$ of the testdata, $R \le 10^{14}.$ Translated by ChatGPT 5