Sums of logarithmic weights involving r-full numbers

Let (n, q) denote the greatest common divisor of positive integers n and q, and let \(f_{r}\) $f_r$ denote the characteristic function of r-full numbers. We consider several asymptotic formulas for sums of the modified square-full ( \(r=2\) $r=2$ ) and cube-full numbers ( \(r=3\) $r=3$ ), which is \(\sum _{n\le y}\sum _{q\le x}\sum _{d|(n,q)}df_{r}\left( \frac{q}{d}\right) \log \frac{x}{q}\) $\sum _{n\le y}^{}\sum _{q\le x}^{}\sum _{d|(n,q)}^{}d{f}_r\left(\frac{q}{d}\right)log\frac{x}{q}$ with any positive real numbers x and y. Moreover, we derive the asymptotic formula of the above with \(r=2\) $r=2$ under the Riemann Hypothesis.