Perfect powers including duplications

Abstract

Let N(x) be the number of perfect powers not exceeding x and let M(x) be the number of perfect powers with duplications not exceeding x. We prove the formulae M(x) ∼ N(x), (M(x)-N(x)) ∼ √ N(x). Let A(x) be the number of duplicated perfect powers not exceeding x. We prove the formula A(x) ∼ √N(x). The number of ways that a duplicated perfect power can be expressed as perfect power we call multiplicity of the perfect power. Let A₂(x) be the number of duplicated perfect powers with multiplicity 2 not exceeding x and let A₃(x) be the number of duplicated perfect powers with multiplicity greater than 2 not exceeding x. We prove the formulae A₂(x) ∼ √N(x), A₃(x) ∼ ∛N(x). We also prove more precise asymptotic formulae for M(x), M(x)-N(x), A(x), A₂(x) and A_3>(x).