Products of consecutive integers with unusual anatomy

Abstract

Call an interval $\{N+1,\dots,N+H\}$ of consecutive natural numbers \emph{bad} if the product $(N+1) \dots (N+H)$ is divisible by the square of its largest prime factor; \emph{very bad} if this product is powerful, and \emph{type $F_3$} if it has the same squarefree component as a factorial. Such concepts arose in the analysis of the factorial equation $a_1! a_2! a_3! = m^2$ with $a_1<a_2<a_3$. Answering several questions of Erd\H{o}s and Graham, we obtain asymptotics for the number of integers contained in bad or very bad intervals, and to get near-asymptotics for the number of right endpoints of a type $F_3$ interval, or on the number of solutions to $a_1! a_2! a_3! = m^2$.