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.,

$$\begin{aligned} {\mathfrak {B}}(x)\asymp \frac{x}{\log x}. \end{aligned}$$

中文翻译：

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复合正整数素数因子之和

令\（{\ mathfrak {B}}（x）\）是直到x的复合正整数的数量，该整数的不同素数和为素数。Luca和Moodley证明存在两个正常数\（a_1 \）和\（a_2 \）使得

$$ \ begin {aligned} a_1x / \ log ^ 3x \ le {\ mathfrak {B}}（x）\ le a_2x / \ log x。\ end {aligned} $$

假设Bateman-Horn猜想的统一形式，他们给出了与上限相同数量级的下限的条件证明。在本文中，我们提供了此结果的无条件证明，即

$$ \ begin {aligned} {\ mathfrak {B}}（x）\ asymp \ frac {x} {\ log x}。\ end {aligned} $$