On Erdős and Sárközy's sequences with Property P

Abstract

A sequence $A$ of positive integers having the property that no element $a_i\in A$ divides the sum $a_j+a_k$ of two larger elements is said to have `Property P'. We construct an infinite set $S\subset \mathbb{N}$ having Property P with counting function $S(x)\gg \frac{\sqrt{x}}{\sqrt{\log x}(\log \log x{)}^2(\log \log \log x{)}^2}$. This improves on an example given by Erdős and Sárközy with a lower bound on the counting function of order $\frac{\sqrt{x}}{\log x}$.

8 pages