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Combinatorica

, Volume 38, Issue 5, pp 1175–1203 | Cite as

Multiplicative Bases and an Erdős Problem

  • Péter Pál Pach
  • Csaba Sándor
Original Paper
  • 34 Downloads

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?

Mathematics Subject Classification (2000)

11B05 11B83 

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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.MathSciNetCrossRefzbMATHGoogle 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.CrossRefzbMATHGoogle Scholar
  3. [3]
    R. E. Dressler: Some new multiplicative bases for the positive integers, Indag. Math. 32 (1970), 338–340.MathSciNetCrossRefzbMATHGoogle 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.zbMATHGoogle Scholar
  5. [5]
    P. P. Pach: Generalized multiplicative Sidon sets, J. Number Theory 157 (2015), 507–529.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    D. Raikov: On multiplicative bases for natural series, Rec. Math. [Mat. Sbornik] N. S. 3 (1938), 569–576.zbMATHGoogle Scholar
  7. [7]
    J. B. Rosser and L. Schoenfeld: Approximate formulas for some functions of prime numbers, Ill. Journ. Math. 6 (1962), 64–94.MathSciNetzbMATHGoogle Scholar
  8. [8]
    P. Turán: On an extremal problem in graph theory, Matematikai és Fizikai Lapok 48 (1941), 436–452 (in Hungarian).Google Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Computer Science and Information TheoryBudapest University of Technology and EconomicsBudapestHungary
  2. 2.Institute of MathematicsBudapest University of Technology and EconomicsBudapestHungary

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