On sums of prime factors

Abstract

We study the arithmetic function sopfr$(n)$ (OEIS A001414) which gives the sum of prime factors (with repetition) of a number $n$. In particular we obtain the asymptotic formula $$\sum _{n\le x}\mathrm{sopfr}(\mathrm{n})\sim \frac{{\pi }^2}{12}\frac{{\mathrm{x}}^2}{\log \mathrm{x}},$$ which holds as well for the function sopf$(n)$ (OEIS A008472) that just gives the sum of distinct prime factors of $n$. This asymptotic formula was already stated by R. Jakimcyuk \cite{rj12} which was brought to our attention after the completion of the first version of this manuscript.

4 pages