Numbers which are only orders of abelian or nilpotent groups

Abstract

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