Number theory
On AP3-covering sequences
Comptes Rendus. Mathématique, Volume 356 (2018) no. 2, pp. 121-124.

Recently, motivated by Stanley's sequences, Kiss, Sándor, and Yang introduced a new type sequence: a sequence A of nonnegative integers is called an APk-covering sequence if there exists an integer n0 such that, if n>n0, then there exist a1A,,ak1A, a1<a2<<ak1<n such that a1,,ak1,n form a k-term arithmetic progression. They prove that there exists an AP3-covering sequence A such that limsupnA(n)/n34. In this note, we prove that there exists an AP3-covering sequence A such that limsupnA(n)/n=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 APk s'il existe un entier n0 tel que, pour tout n>n0, il existe a1,,ak1A, a1<a2<<ak1<n tels que a1,,ak1,n soit une progression arithmétique à k termes. Ils démontrent qu'il existe une suite d'entiers A qui est AP3 et satisfait limsupnA(n)/n34. Nous montrons ici qu'il en existe une satisfaisant limsupnA(n)/n=15.

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DOI: 10.1016/j.crma.2017.12.013
Chen, Yong-Gao 1

1 School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, PR China
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Chen, Yong-Gao. On AP3-covering sequences. Comptes Rendus. Mathématique, Volume 356 (2018) no. 2, pp. 121-124. doi : 10.1016/j.crma.2017.12.013. http://www.numdam.org/articles/10.1016/j.crma.2017.12.013/

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

[5] Moy, R.A. 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.

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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.