A Generalization of Maximal $k$-Multiple-Free Sets of Integers

Abstract

Let $k$ and $b$ be integers and $k>1$. A set $S$ of integers is called $(k,b)$ linear-free (or $(k,b)$-LF for short) if $2\in S$ implies $kx+b\notin S$. Let $F(n,k,b)=max\{|A|:A\text{ is }(k,0)\text{-LF and }A\subseteq [1,n]\}$, where $[1,n]$ denotes all integers between $1$ and $n$. A subset $A$ of $[1,n]$ with $|A|=F(n,k,b)$ is called a maximal $(k,b)$-LF subset of $[1,n]$. In this paper, a recurrence relation for $F(n,k,b)$ is obtained and a method to construct a maximal $(k,b)$-LF subset of $[1,n]$ is given.