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.,

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

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Alladi, K., Erdös, P.: On an additive arithmetic function. Pacific J. Math. 71, 275–295 (1977)

  2. 1.

    Bateman, P.T., Horn, R.A.: A heuristic asympyotic formula considering the distribution of prime numbers. Math. Comput. 16, 363–367 (1962)

    Article  Google Scholar 

  3. 2.

    De Koninck, F.Luca: Integers divisible by the sum of their prime factors. Mathematika 52, 69–77 (2005)

    MathSciNet  Article  Google Scholar 

  4. 3.

    de Polignac, A.: Six propositions arithmologiques déduites du crible d’Erathosténe. Nouv. Ann. Math. 8, 423–429 (1849)

    Google Scholar 

  5. 4.

    Erdős, P., Pomerance, C.: On the largest prime factor of \(n\) and \(n+1\). Aequationes Math. 17, 311–321 (1978)

    MathSciNet  Article  Google Scholar 

  6. 5.

    Huxley, M.N.: On the difference between consecutive primes. Invent. Math. 15, 155–164 (1972)

    MATH  Google Scholar 

  7. 6.

    Luca, F., Moodley, D.: Composite positive integers whose sum of prime factors is prime. Arch. Math. 56, 49–64 (2020)

    MathSciNet  MATH  Google Scholar 

  8. 7.

    Nelson, C., Penney, D.E., Pomerance, C.: 714 and 715. J. Recreational Math. 7, 87–89 (1974)

    MathSciNet  Google Scholar 

  9. 8.

    Perelli, A., Pintz, J.: On the exceptional set for the \(2k\)-twin primes problem. Compositio Math. 82, 355–372 (1992)

    MathSciNet  MATH  Google Scholar 

  10. 9.

    Vaughan, R.C., Weis, K.L.: On sigma-phi numbers. Mathematika 48, 169–189 (2001)

    MathSciNet  Article  Google Scholar 

  11. 10.

    Wang, Z.: Sur les plus grands facteurs premiers d’entiers consécutifs (French). Mathematika 64, 343–379 (2018)

    MathSciNet  Article  Google Scholar 

  12. 11.

    Wang, Z.: Autour des plus grands facteurs premiers d’entiers consécutifs voisins d’un entier criblé (French). Q. J. Math. 69, 995–1013 (2018)

    MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Yuchen Ding.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

Keywords

  • Prime factors
  • Primes
  • Twin primes
  • Exceptional sets

Mathematics Subject Classification

  • 11N25
  • 11A41