On sum of prime factors of composite positive integers


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.,

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

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Correspondence to Yuchen Ding.

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Yuchen Ding is supported by the National Natural Science Foundation of China (Grant No. 11971224). Xiaodong Lü is supported by the Natural Science Foundation of Higher Education Institutions of Jiangsu (No. 18KJB110032)

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Ding, Y., Lü, X. On sum of prime factors of composite positive integers. Ramanujan J (2021). https://doi.org/10.1007/s11139-020-00370-y

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  • Prime factors
  • Primes
  • Twin primes
  • Exceptional sets

Mathematics Subject Classification

  • 11N25
  • 11A41