Counting Of Coprime And Square-free Triples

Coprime and square-free number are two elementary concepts in number theory. We have studied the counting of coprime square-free numbers in this paper. For three arbitrarily and independently selected integers not exceeding x, Let A₁(x) denote the number of triples (a, b, c), where a, b, c are pairwise coprime and a is square-free.Let A₃(x) denote the number of triples (a, b, c) if a, b, c are pairwise coprime and are all square-free. We give two asymptotic formulas about A₁(x) and A₃(x) in this paper.