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

Let \(k\) be an integer greater than one. A set \(S\) of integers is called \(k\)-multiple-free (or \(k\)-MF for short) if \(x \in S\) implies \(kx \notin S\). Let \(f_k(n) = \max\{|A| : A \subseteq [1,n] \text{ is } k\text{-MF}\}\). A subset \(A\) of \([1,n]\) with \(|A| = f_k(n)\) is called a maximal \(k\)-MF subset of \([1,n]\). In this paper, we give a recurrence relation and a formula for \(f_k(n)\). In addition, we give a method for constructing a maximal \(k\)-MF subset of \([1,n]\).