Multiplicative Bases and an Erdős Problem

Combinatorica volume 38, pages1175–1203(2018)Cite this article

Abstract

In this paper we investigate how small the density of a multiplicative basis of order h can be in {1,2,...,n} and in ℤ+. Furthermore, a related problem of Erdős is also studied: How dense can a set of integers be, if none of them divides the product of h others?

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References

  1. [1]

    T. H. Chan: On sets of integers, none of which divides the product of k others, European J. Comb. 32 (2011), 443–447.

    MathSciNet  Article  MATH  Google Scholar 

  2. [2]

    T. H. Chan, E. Győri and A. Sárközy: On a problem of Erdős on integers, none of which divides the product of k others, European J. Comb. 31 (2010), 260–269.

    Article  MATH  Google Scholar 

  3. [3]

    R. E. Dressler: Some new multiplicative bases for the positive integers, Indag. Math. 32 (1970), 338–340.

    MathSciNet  Article  MATH  Google Scholar 

  4. [4]

    P. Erdős: On sequences of integers no one of which divides the product of two others and on some related problems, Tomsk. Gos. Univ. Uchen. Zap 2 (1938), 74–82.

    MATH  Google Scholar 

  5. [5]

    P. P. Pach: Generalized multiplicative Sidon sets, J. Number Theory 157 (2015), 507–529.

    MathSciNet  Article  MATH  Google Scholar 

  6. [6]

    D. Raikov: On multiplicative bases for natural series, Rec. Math. [Mat. Sbornik] N. S. 3 (1938), 569–576.

    MATH  Google Scholar 

  7. [7]

    J. B. Rosser and L. Schoenfeld: Approximate formulas for some functions of prime numbers, Ill. Journ. Math. 6 (1962), 64–94.

    MathSciNet  MATH  Google Scholar 

  8. [8]

    P. Turán: On an extremal problem in graph theory, Matematikai és Fizikai Lapok 48 (1941), 436–452 (in Hungarian).

    Google Scholar 

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Author information

Affiliations

  1. Department of Computer Science and Information Theory, Budapest University of Technology and Economics, 1117, Budapest, Hungary

    Péter Pál Pach

  2. Institute of Mathematics, Budapest University of Technology and Economics, H-1529, P.O. Box, Budapest, Hungary

    Csaba Sándor

Authors
  1. Péter Pál Pach
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  2. Csaba Sándor
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Corresponding author

Correspondence to Csaba Sándor.

Additional information

This author was supported by the Hungarian Scientific Research Funds (Grant Nr. OTKA PD115978 and OTKA K108947) and the János Bolyai Research Scholarship of the Hungarian Academy of Sciences.

This author was supported by the OTKA Grant No. K109789 and the János Bolyai Research Scholarship of the Hungarian Academy of Sciences.

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Pach, P.P., Sándor, C. Multiplicative Bases and an Erdős Problem. Combinatorica 38, 1175–1203 (2018). https://doi.org/10.1007/s00493-016-3588-2

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Mathematics Subject Classification (2000)

  • 11B05
  • 11B83