On the index of friability

Given an integer [Formula: see text], let [Formula: see text] stand for its largest prime factor, setting [Formula: see text]. We introduce the arithmetic function [Formula: see text] (with [Formula: see text]) and call it the index of friability of the integer n. The index of friability of an integer is an absolute measure of its friability (or smoothness). We first determine the respective mean values of the functions [Formula: see text] and [Formula: see text], thereafter obtaining various estimates comparing the index of friability with the index of composition. Then, given any finite set of natural numbers, we order its members according to their index of friability and obtain results regarding their distribution. In particular, this allows us to construct arbitrarily long monotonic sequences of integers with increasing index of friability.