Let \(n=p_1^{\alpha_1}\cdots p_k^{\alpha_k}\). Given fixed integers \(h,r\geq 2\), we say that \(n\) is a \(r\)-free number if \(\max\{\alpha_1,\dots,\alpha_k\}\leq r-1\), and we say that \(n\) is a \(h\)-full number if \(\min\{\alpha_1,\dots,\alpha_k\}\geq h\). In the paper under review, the authors provide asymptotic expansions for the sums \[ \sum_{\substack{n\leq x\\ \text{\(n\) is \(r\)-free}}}P(n)\quad \text{and}\quad \sum_{\substack{n\leq x\\ \text{\(n\) is \(h\)-full}}}P(n), \] where \(P(n)\) is the largest prime factor of \(n\), with \(P(1) = 1\). More precisely, they prove that for any integer \(m\geq 1\) there exist constants \(d_1,\dots,d_m\) and \(e_1,\dots,e_m\) such that \[ \sum_{\substack{n\leq x\\ \text{\(n\) is \(r\)-free}}}P(n)=x^2\sum_{j=1}^m\frac{d_j}{\log^j x}+O\left(\frac{x^2}{\log^{m+1}x}\right), \] and \[ \sum_{\substack{n\leq x\\ \text{\(n\) is \(h\)-full}}}P(n)=x^{2/h}\sum_{j=1}^m\frac{e_j}{\log^j x}+O\left(\frac{x^{2/h}}{\log^{m+1}x}\right). \]