On certain large additive functions

Let P(n) denote the largest prime factor of $n\ge 2,P(1)=1$, and let $$\beta (n)=\sum _{p\mathrm{\mid }n}p,\mathrm{B}(n)=\sum _{p^{\alpha }\mathrm{\mid }\mathrm{\mid }n}\alpha p,{\mathrm{B}}_1(n)=\sum _{{\text{p}}^{\alpha }\mathrm{\mid }\mathrm{\mid }n}{p}^{\alpha }$$ denote “large” additive functions. A survey of results on the functions $P(n),\beta (n),\mathrm{B}(n)$ and ${\mathrm{B}}_1(n)$ is presented, as well as some new results and open problems.