Summary: Let $A$ be a finite union of disjoint sets of consecutive integers and let $n$ be a positive integer. We give a formula for the number of relatively prime subsets (resp., relatively prime subsets of cardinality $k$) of $A$, which generalizes results of Nathanson, El Bachraoui and others. We give as well similar formulas for the number of subsets with gcd coprime to $n$.

phi function, relatively prime set, combinatorial identity