A Generalization of Maximal \(k\)-Multiple-Free Sets of Integers

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.