On sum of prime factors of composite positive integers

Abstract

Let $${\mathfrak {B}}(x)$$ be the number of composite positive integers up to x whose sum of distinct prime factors is a prime number. Luca and Moodley proved that there exist two positive constants $$a_1$$ and $$a_2$$ such that $$\begin{aligned} a_1x/\log ^3x\le {\mathfrak {B}}(x)\le a_2x/\log x. \end{aligned}$$ Assuming a uniform version of the Bateman–Horn conjecture, they gave a conditional proof of a lower bound of the same order of magnitude as the upper bound. In this paper, we offer an unconditional proof of the this result, i.e., $$\begin{aligned} {\mathfrak {B}}(x)\asymp \frac{x}{\log x}. \end{aligned}$$