Let p(n) denote the largest prime factor of an integer \(n\geq 2\), and \(p(1)=1\), and let \(d_ k(n)\) be the divisor function. Asymptotic formulas for the sums \(\sum_{n\leq x,p(n)\leq y}d_ k(n)\) and \(\sum_{n\leq x}d_ k(n)/p(n)\) are derived, respectively. The estimates are made to depend on \(\Psi\) (x,y), the number of integers not exceeding x, all of whose prime factors do not exceed y, and on \(\rho\) (u), the Dickman function.