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2013 | OriginalPaper | Buchkapitel

On Divisibility Properties of Sequences of Integers

verfasst von : András Sárközy

Erschienen in: The Mathematics of Paul Erdős I

Verlag: Springer New York

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Abstract

Our first joint paper with Erdős appeared in 1966. It was a triple paper with Szemerédi written on divisibility properties of sequences of integers which is one of Erdős’ favorite subjects. Nine further triple papers written on the same subject followed it, and since 1966, we have written altogether 52 joint papers with Erdős. On this special occasion I would like to return to the subject of our very first paper. In Sect. 2, I will give a survey of the related results, while in Sect. 3, I will study a further related problem.

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Literatur
1.
Zurück zum Zitat R. Alexander, Density and multiplicative structure of sets of integers, Acta Arith. 12 (1966/67), 321–332. R. Alexander, Density and multiplicative structure of sets of integers, Acta Arith. 12 (1966/67), 321–332.
3.
5.
Zurück zum Zitat I. Anderson, S. D. Cohen and W.W. We Stothers, Primitive polynomial sequences, Mathematika 21 (1974), 239–247.MathSciNetCrossRef I. Anderson, S. D. Cohen and W.W. We Stothers, Primitive polynomial sequences, Mathematika 21 (1974), 239–247.MathSciNetCrossRef
6.
Zurück zum Zitat F. Behrend, On sequences of numbers not divisible one by another, J. London Math. Soc. 10 (1935), 42–44. F. Behrend, On sequences of numbers not divisible one by another, J. London Math. Soc. 10 (1935), 42–44.
8.
Zurück zum Zitat H. Davenport and P. Erdős, On sequences of positive integers, Acta Arith. 2 (1937), 147–151 and J. Indian Math. Soc. 15 (1951), 19–24. H. Davenport and P. Erdős, On sequences of positive integers, Acta Arith. 2 (1937), 147–151 and J. Indian Math. Soc. 15 (1951), 19–24.
9.
Zurück zum Zitat P. Erdős, Note on sequences of integers no one of which is divisible by any other, J. London Math. Soc. 10 (1935), 126–128.CrossRef P. Erdős, Note on sequences of integers no one of which is divisible by any other, J. London Math. Soc. 10 (1935), 126–128.CrossRef
10.
Zurück zum Zitat P. Erdős, Integers with exactly k prime factors, Ann. Math. 49 (1948), 53–66.CrossRef P. Erdős, Integers with exactly k prime factors, Ann. Math. 49 (1948), 53–66.CrossRef
11.
Zurück zum Zitat P. Erdős, Some extremal problems on divisibility properties of sequences of integers, Fibonacci Quart. 19 (1981), 208–213.MathSciNet P. Erdős, Some extremal problems on divisibility properties of sequences of integers, Fibonacci Quart. 19 (1981), 208–213.MathSciNet
12.
Zurück zum Zitat P. Erdős and A. Sárközy, On the divisibility properties of sequences of integers, Proc. London Math. Soc. (3) 21 (1970), 97–101. P. Erdős and A. Sárközy, On the divisibility properties of sequences of integers, Proc. London Math. Soc. (3) 21 (1970), 97–101.
13.
Zurück zum Zitat P. Erdős and A. Sárközy, Some asymptotic formulas on generalized divisor functions, I, Studies in Pure Mathematics, To the Memory of Paul Turán, Akadémiai Kiadó, 1983, 165–179. P. Erdős and A. Sárközy, Some asymptotic formulas on generalized divisor functions, I, Studies in Pure Mathematics, To the Memory of Paul Turán, Akadémiai Kiadó, 1983, 165–179.
14.
Zurück zum Zitat P. Erdős and A. Sárközy, Some asymptotic formulas on generalized divisor functions, II, J. Number Theory 15 (1982), 115–136.MathSciNetCrossRef P. Erdős and A. Sárközy, Some asymptotic formulas on generalized divisor functions, II, J. Number Theory 15 (1982), 115–136.MathSciNetCrossRef
15.
Zurück zum Zitat P. Erdős and A. Sárközy, Some asymptotic formulas on generalized divisor functions, III, Acta Arith. 41 (1982), 395–411.MathSciNet P. Erdős and A. Sárközy, Some asymptotic formulas on generalized divisor functions, III, Acta Arith. 41 (1982), 395–411.MathSciNet
16.
Zurück zum Zitat P. Erdős and A. Sárközy, Some asymptotic formulas on generalized divisor functions, IV, Studia Sci. Math. Hung. 15 (1980), 467–479. P. Erdős and A. Sárközy, Some asymptotic formulas on generalized divisor functions, IV, Studia Sci. Math. Hung. 15 (1980), 467–479.
17.
Zurück zum Zitat P. Erdős, A. Sárközy and E. Szemerédi, On divisibility properties of sequences of integers, Studia Sci. Math, Hung. 1 (1966), 431–435. P. Erdős, A. Sárközy and E. Szemerédi, On divisibility properties of sequences of integers, Studia Sci. Math, Hung. 1 (1966), 431–435.
18.
Zurück zum Zitat P. Erdős, A. Sárközy and E. Szemerédi, On the divisibility properties of sequences of integers, I, Acta Arith. 11 (1966), 411–418.MathSciNet P. Erdős, A. Sárközy and E. Szemerédi, On the divisibility properties of sequences of integers, I, Acta Arith. 11 (1966), 411–418.MathSciNet
19.
Zurück zum Zitat P. Erdős, A. Sárközy and E. Szemerédi, On the divisibility properties of sequences of integers, II, 14 (1967/68), 1–12. P. Erdős, A. Sárközy and E. Szemerédi, On the divisibility properties of sequences of integers, II, 14 (1967/68), 1–12.
20.
Zurück zum Zitat P. Erdős, A. Sárközy and E. Szemerédi, On an extremal problem concerning primitive sequences, J. London Math. Soc. 42 (1967), 484–488.MathSciNetCrossRef P. Erdős, A. Sárközy and E. Szemerédi, On an extremal problem concerning primitive sequences, J. London Math. Soc. 42 (1967), 484–488.MathSciNetCrossRef
21.
Zurück zum Zitat P. Erdős, A. Sárközy and E. Szemerédi, On a theorem of Behrend, J. Australian Math. Soc. 7 (1967), 9–16.CrossRef P. Erdős, A. Sárközy and E. Szemerédi, On a theorem of Behrend, J. Australian Math. Soc. 7 (1967), 9–16.CrossRef
22.
Zurück zum Zitat P. Erdős, A. Sárközy and E. Szemerédi, On divisibility properties of sequences of integers, Number Theory, Coll. Math. Soc. J. Bolyai 2 (1970), 35–49. P. Erdős, A. Sárközy and E. Szemerédi, On divisibility properties of sequences of integers, Number Theory, Coll. Math. Soc. J. Bolyai 2 (1970), 35–49.
23.
Zurück zum Zitat P. Erdős and Zhang Zhenxiang, Upper bound of ∑1 ∕ (a i loga i ) for primitive sequences, Proc. Amer. Math. Soc., to appear. P. Erdős and Zhang Zhenxiang, Upper bound of 1 ∕ (a i loga i ) for primitive sequences, Proc. Amer. Math. Soc., to appear.
24.
Zurück zum Zitat J. A. Haight, On multiples of certain real sequences, Acta Arith. 49 (1988), 302–306.MathSciNet J. A. Haight, On multiples of certain real sequences, Acta Arith. 49 (1988), 302–306.MathSciNet
25.
Zurück zum Zitat W. Klotz, Generalization of some theorems on sets of multiples and primitive sequences, Acta Arith. 32 (1977), 15–26.MathSciNetMATH W. Klotz, Generalization of some theorems on sets of multiples and primitive sequences, Acta Arith. 32 (1977), 15–26.MathSciNetMATH
26.
Zurück zum Zitat H. G. Meijer, On the upper asymptotic density of (0, r)-primitive sequences, Acta Arith. 25 (1973/74), 191–197. H. G. Meijer, On the upper asymptotic density of (0, r)-primitive sequences, Acta Arith. 25 (1973/74), 191–197.
27.
Zurück zum Zitat H. G. Meijer, Note on (0, r)-primitive sequences, Delft Progress Rep. Ser. F1 (1974/75), no. 3, 82–84. H. G. Meijer, Note on (0, r)-primitive sequences, Delft Progress Rep. Ser. F1 (1974/75), no. 3, 82–84.
28.
Zurück zum Zitat S. Pillai, On numbers which are not multiples of any other in the set, Proc. Indian Acad. Sci. A 10 (1939), 392–394.MathSciNet S. Pillai, On numbers which are not multiples of any other in the set, Proc. Indian Acad. Sci. A 10 (1939), 392–394.MathSciNet
29.
Zurück zum Zitat C. Pomerance and A. Sárközy, On homogeneous multiplicative hybrid problems in number theory, Acta Arithmetica 49 (1988), 291–302.MathSciNetMATH C. Pomerance and A. Sárközy, On homogeneous multiplicative hybrid problems in number theory, Acta Arithmetica 49 (1988), 291–302.MathSciNetMATH
30.
Zurück zum Zitat R. Sattler, A theorem concerning \((r_{1},r_{2},\ldots,r_{t})\)-primitive sequences and an application to (0, r)-primitive sequences, Delft Progress Rep. 2 (1976/ 77), no. 1, 23–26. R. Sattler, A theorem concerning \((r_{1},r_{2},\ldots,r_{t})\)-primitive sequences and an application to (0, r)-primitive sequences, Delft Progress Rep. 2 (1976/ 77), no. 1, 23–26.
31.
Zurück zum Zitat R. Ahlswede, L. H. Khachatrian and A. Sárközy, On the quotient sequence of sequences of integers, Acta Arith. 91 (1999), 117–132.MathSciNetMATH R. Ahlswede, L. H. Khachatrian and A. Sárközy, On the quotient sequence of sequences of integers, Acta Arith. 91 (1999), 117–132.MathSciNetMATH
32.
Zurück zum Zitat R. Ahlswede, L. H. Khachatrian and A. Sárközy, On the counting function of primitive sets of integers, J. Number Theory 79 (1999), 330–341.MathSciNetMATHCrossRef R. Ahlswede, L. H. Khachatrian and A. Sárközy, On the counting function of primitive sets of integers, J. Number Theory 79 (1999), 330–341.MathSciNetMATHCrossRef
33.
Zurück zum Zitat R. Ahlswede, L. H. Khachatrian and A. Sárközy, On prefix-free and suffix-free sequences of integers, in: Numbers, Information and Complexity, eds. I Althöfer et al., Kluwer Academic Publishers, Boston, 2000; 1–16. R. Ahlswede, L. H. Khachatrian and A. Sárközy, On prefix-free and suffix-free sequences of integers, in: Numbers, Information and Complexity, eds. I Althöfer et al., Kluwer Academic Publishers, Boston, 2000; 1–16.
34.
Zurück zum Zitat R. Ahlswede, L. Khachatrian and A. Sárközy, On primitive sets of squarefree integers, Periodica Math. Hungar. 42 (2001), 99–115.MATHCrossRef R. Ahlswede, L. Khachatrian and A. Sárközy, On primitive sets of squarefree integers, Periodica Math. Hungar. 42 (2001), 99–115.MATHCrossRef
35.
36.
Zurück zum Zitat M. Beiglböck, V. Bergelson, N. Hindman and D. Straus, Multiplicative structures in additively large sets, J. Combin. Theory Ser. A 113 (2006), 1219–1242.MathSciNetMATHCrossRef M. Beiglböck, V. Bergelson, N. Hindman and D. Straus, Multiplicative structures in additively large sets, J. Combin. Theory Ser. A 113 (2006), 1219–1242.MathSciNetMATHCrossRef
37.
38.
Zurück zum Zitat T. H. Chan, E. Győri and A. Sárközy, On a problem of Erdős on integers, non of which divides the product of k others, European J. Combin. 31 (2010), 260–269MathSciNetMATHCrossRef T. H. Chan, E. Győri and A. Sárközy, On a problem of Erdős on integers, non of which divides the product of k others, European J. Combin. 31 (2010), 260–269MathSciNetMATHCrossRef
39.
Zurück zum Zitat 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. 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.
40.
41.
Zurück zum Zitat S. Porubsky, Primitive sequences in arithmetical semigroups, Tatra Mt. Math. Publ. 32 (2005), 85–101.MathSciNetMATH S. Porubsky, Primitive sequences in arithmetical semigroups, Tatra Mt. Math. Publ. 32 (2005), 85–101.MathSciNetMATH
Metadaten
Titel
On Divisibility Properties of Sequences of Integers
verfasst von
András Sárközy
Copyright-Jahr
2013
Verlag
Springer New York
DOI
https://doi.org/10.1007/978-1-4614-7258-2_15