Let $\beta(n)=\sum_{p\mid n}p$ and $B(n)=\sum_{p^{\alpha}\parallel n}\alpha p$. Let $p(n)$ denote the largest prime factor of an integer $n\ge2$. In the present paper we sharpen the asymptotic formula for the sum $\sum\limits_{2\le n\le x} B(n)/\beta(n)$ and we derive an asymptotic formula for the sum $\sum\limits_{2\le n\le x}(B(n)-\beta(n))/p(n)$.