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On Prefix-Free and Suffix-Free Sequences of Integers

  • Rudolf Ahlswede
  • Levon H. Khachatrian
  • András Sárközy
Chapter

Abstract

The set of the positive integers and positive square—free integers are denoted by IN and IN*, respectively, and we write IN(n) = IN ∩ [1, n], IN* (n) = IN* ∩ [1, n], where [1, n] = {1, 2, ... , n}. The set of primes is denoted by P. The smallest and greatest prime factors of the positive integer n are denoted by p(n) and P(n), respectively. ω(n) denotes the number of distinct prime factors of n, while Ω(n) denotes the number of prime factors of n counted with multiplicity:
$$\omega \left( n \right)\, = \,\sum\limits_{p|n} {1,} \,\Omega \left( n \right)\, = \,\sum\limits_{{p^\alpha }\parallel n} {\alpha .} $$
µ(n) denotes the Möbius function.

Keywords

Counting Function Multiplicative Function Basic Lemma Logarithmic Density Divisibility Property 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    R. Ahlswede, L. Khachatrian and A. Sârközy, “On the counting function of primitive sets of integers”, Preprint 98–077, SFB 343Diskrete Strukturen in der Mathematik”,Universität Bielefeld, submitted to J. Number Theory.Google Scholar
  2. [2]
    R. Ahlswede and L.H. Khachatrian, “Classical results on primitive and recent results on cross-primitive sequences”, in: The Mathematics of Paul Erdös, vol. I, eds.R.L. Graham and J. Nesetril, Algorithms and Combinatorics 13, Springer-Verlag, 1997; 104–116.Google Scholar
  3. [3]
    A.S. Besicovitch, “On the density of certain sequences“, Math. Ann. 110, 1934, 336–341.MathSciNetCrossRefGoogle Scholar
  4. [4]
    F. Behrend, “On sequences of numbers not divisible by one another”, J. London Math. Soc., 10, 1935, 42–44.CrossRefGoogle Scholar
  5. [5]
    P. Erdös, “Note on sequences of integers no one of which is divisible by any other”, J. London Math. Soc., 10, 1935, 126–128.CrossRefGoogle Scholar
  6. [6]
    P. Erdös, “A generalization of a theorem of Besicovitch”, J. London Math. Soc., 11, 1935, 92–98.CrossRefGoogle Scholar
  7. [7]
    P. Erdös, A. Sârközy and E. Szemerédi, “On a theorem of Behrend“, J. Australian Math. Soc., 7, 1967, 9–16.zbMATHCrossRefGoogle Scholar
  8. [8]
    P. Erdös, A. Sârközy and E. Szemerédi, “On divisibility properties of sequences of integers”, Coll. Math. Soc. J. Bolyai, 2, 1970, 35–49.Google Scholar
  9. [9]
    H. Halberstam and K.F. Roth, “Sequences”, Springer-Verlag, BerlinHeidelberg-New York, 1983.zbMATHCrossRefGoogle Scholar
  10. [10]
    A. Sârközy, “On divisibility properties of sequences of integers”, in: The Mathematics of Paul Erdös, eds. R.L. Graham and J. Nesetril, Algorithms and Combinatorics 13, Springer-Verlag, 1997, 241–250.Google Scholar
  11. [11]
    A. Selberg, “Note on a paper by L.G. Sathe”, J. Indian Math. Soc., 18, 1954, 83–87.MathSciNetzbMATHGoogle Scholar
  12. [12]
    H. Davenport and P. Erdös, “On sequences of positive integers”, Acta Arith., 2, 1936, 147–151.Google Scholar
  13. [13]
    G.H. Hardy and S. Ramanujan, “The normal number of prime factors of a number n”, Quarterly J. Math., 48, 1920, 76–92.Google Scholar

Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • Rudolf Ahlswede
    • 1
  • Levon H. Khachatrian
    • 1
  • András Sárközy
    • 2
  1. 1.Fakultät für MathematikUniversität BielefeldBielefeldGermany
  2. 2.Department of Algebra and Number TheoryEötvös UniversityBudapestHungary

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