On the distribution of powerful and r-free lattice points

Let $1 < c < 2$. For m, n ∈ N, a lattice point $(m, n)$ is powerful if and only if (m, n) is a powerful number, where (*, *) is the greatest common divisor function. In this paper, we count the number of the ordered pairs $(m, n)$, m, n ≤ x such that the lattice point ( mᶜ , nᶜ ) is powerful. Moreover, we study r-free lattice points analogues of powerful lattice points.