A generalization of maximal \(k\)-multiple-free sets of integers
The authors consider the quantity \(F(n,k,b)\) that is defined as the maximum size of a subset \(X\) of \(\{1,2,\ldots ,n\}\) with the property that \(x\in X\) implies \(kx+b\not \in X\) (\(n,k\) are positive integers and \(b\) is an integer). They prove that \(F(n,k,b)=n\) for \(n<k+b\) and, if \(k-1+b\neq 0\), NEWLINE\[NEWLINEF(n,k,b)=n-\lfloor (n-b)/k\rfloor + F(\lfloor (\lfloor (n-b)/k\rfloor -b)/k\rfloor ,k,b).NEWLINE\]NEWLINE \(X\) with the maximum cardinality is described.
scientific article; zbMATH DE number 1682821