On sets of integers, none of which divides the product of \(k\) others
type: Item
instance of: scholarly article
title: On sets of integers, none of which divides the product of \(k\) others
zbMATH Open document ID: 1230.11025
author: Tsz Ho Chan
published in: European Journal of Combinatorics
publication date: 2011-03-04T00:00:00Z
review text: For an integer \(k\geq 1\) let \(F_k(x)\) be the maximal number of elements of a set \({\mathcal A}\subset {\mathbb N}\cap [1,x]\) such that there are no distinct elements \(a,a_1,\dots,a_k\in {\mathcal A}\) with \(a\) dividing \(a_1a_2\dots a_k\). The author proves that there exist absolute constants \( c_1,c_2>0\) such that, for any real number \(x>e^{48}\) and any integer \(2\leq k\leq \frac 16\sqrt{\frac{\log x}{\log\log x}}\), \[ \pi(x)+\frac{c_2}{(k+1)^2}\frac{x^{\frac{2}{k+1}}}{(\log x)^2}\leq F_k(x)\leq \pi(x)+c_1(k+1)^2\frac{x^{\frac{2}{k+1}}}{(\log x)^2}. \]
reviewed by: Florin Nicolae
Mathematics Subject Classification ID: 11B75
zbMATH DE Number: 5860465
zbMATH Keywords: sets of integers
zbMATH Keywords: divisibility
MaRDI profile type: Publication
full work available at URL: j.ejc.2010.11.010
OpenAlex ID: W1971502753
cites work: On a problem of Erdős on integers, none of which divides the product of \(k\) others
DOI: 10.1016/J.EJC.2010.11.010
description: scientific article; zbMATH DE number 5860465