On near relative prime number in a sequence of positive integers

Abstract

Let $A$ be a sequence of positive integers. An element $\alpha$ of $A$ is called an {\it s-near relative prime number} ($s$-near relprime in short) if $\alpha$ is coprime to any distinct element in $A$ except exactly $s$ elements of $A$. In this paper, we study the existence of an arithmetic sequence with no 1-near relprimes.