On AP3-covering sequences

Yong-Gao Chen

doi:10.1016/j.crma.2017.12.013

Number theory

On AP₃-covering sequences

Presented by: the Editorial Board

Yong-Gao Chen ¹

¹ School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, PR China

Comptes Rendus. Mathématique, Volume 356 (2018) no. 2, pp. 121-124.

Abstract
Résumé

Recently, motivated by Stanley's sequences, Kiss, Sándor, and Yang introduced a new type sequence: a sequence A of nonnegative integers is called an $A P_{k}$ -covering sequence if there exists an integer $n_{0}$ such that, if $n > n_{0}$ , then there exist $a_{1} \in A, \dots, a_{k - 1} \in A$ , $a_{1} < a_{2} < \dots < a_{k - 1} < n$ such that $a_{1}, \dots, a_{k - 1}, n$ form a k-term arithmetic progression. They prove that there exists an $A P_{3}$ -covering sequence A such that $\underset{n \to \infty}{\lim \sup} A (n) / \sqrt{n} \leq 34$ . In this note, we prove that there exists an $A P_{3}$ -covering sequence A such that $\underset{n \to \infty}{\lim \sup} A (n) / \sqrt{n} = \sqrt{15}$ .

Motivés par la définition des suites de Stanley, Kiss, Sándor et Yang ont récemment introduit un nouveau type de suites : une suite d'entiers positifs ou nuls A est dite $A P_{k}$ s'il existe un entier $n_{0}$ tel que, pour tout $n > n_{0}$ , il existe $a_{1}, \dots, a_{k - 1} \in A$ , $a_{1} < a_{2} < \dots < a_{k - 1} < n$ tels que $a_{1}, \dots, a_{k - 1}, n$ soit une progression arithmétique à k termes. Ils démontrent qu'il existe une suite d'entiers A qui est $A P_{3}$ et satisfait $\lim \sup_{n \to \infty} A (n) / \sqrt{n} \leq 34$ . Nous montrons ici qu'il en existe une satisfaisant $\lim \sup_{n \to \infty} A (n) / \sqrt{n} = \sqrt{15}$ .

Received: 2017-10-31
Accepted: 2017-12-22
Published online: 2018-01-11

DOI: 10.1016/j.crma.2017.12.013

Author's affiliations:

Yong-Gao Chen ¹

¹ School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, PR China

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     author = {Yong-Gao Chen},
     title = {On {\protect\emph{A}\protect\emph{P}\protect\textsubscript{3}-covering} sequences},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {121--124},
     publisher = {Elsevier},
     volume = {356},
     number = {2},
     year = {2018},
     doi = {10.1016/j.crma.2017.12.013},
     language = {en},
}

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Yong-Gao Chen. On AP₃-covering sequences. Comptes Rendus. Mathématique, Volume 356 (2018) no. 2, pp. 121-124. doi : 10.1016/j.crma.2017.12.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.12.013/

References
Cited by

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[2] P. Erdős; V. Lev; G. Rauzy; C. Sándor; A. Sárközy Greedy algorithm, arithmetic progressions, subset sums and divisibility, Discrete Math., Volume 200 (1999), pp. 119-135

[3] J. Gerver; L.T. Ramsey Sets of integers with no long arithmetic progressions generated by the greedy algorithm, Math. Comput., Volume 33 (1979), pp. 1353-1359

[4] S.Z. Kiss; C. Sándor; Q.-H. Yang On generalized Stanley sequences | arXiv

[5] R.A. Moy On the growth of the counting function of Stanley sequences, Discrete Math., Volume 311 (2011), pp. 560-562

[6] A.M. Odlyzko, R.P. Stanley, Some curious sequences constructed with the greedy algorithm, Bell Laboratories internal memorandum, 1978.

Cited by Sources:

^☆ The work is supported by the National Natural Science Foundation of China, Grant No. 11771211 and by a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

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