PMO 2022 Problem 3

Call a lattice point visible if the line segment connecting the point and the origin does not pass through another lattice point. Given a positive integer $k$ , denote by $S_k$ the set of all visible lattice points $(x, y)$ such that $x^2 + y^2 = k^2$ . Let $D$ denote the set of all positive divisors of $2021 \cdot 2025$ . Compute the sum $$\sum_{d \in D} |S_d|$$ Here, a lattice point is a point $(x, y)$ on the plane where both $x$ and $y$ are integers, and $|A|$ denotes the number of elements of the set $A$ .