Numbers which are only orders of abelian or nilpotent groups

Abstract

Refining a result of Erdős and Mays, we give asymptotic series expansions for the functions A(x)-C(x), the count of n≤x for which every group of order n is abelian (but not all cyclic), and N(x)-A(x), the count of n≤x for which every group of order n is nilpotent (but not all abelian).