A generalization of the practical numbers
A positive integer [Formula: see text] is practical if every [Formula: see text] can be written as a sum of distinct divisors of [Formula: see text]. One can generalize the concept of practical numbers by applying an arithmetic function [Formula: see text] to each of the divisors of [Formula: see text] and asking whether all integers in a certain interval can be expressed as sums of [Formula: see text]’s, where the [Formula: see text]’s are distinct divisors of [Formula: see text]. We will refer to such [Formula: see text] as “[Formula: see text]-practical”. In this paper, we introduce the [Formula: see text]-practical numbers for the first time. We give criteria for when all [Formula: see text]-practical numbers can be constructed via a simple necessary-and-sufficient condition, demonstrate that it is possible to construct [Formula: see text]-practical sets with any asymptotic density, and prove a series of results related to the distribution of [Formula: see text]-practical numbers for many well-known arithmetic functions [Formula: see text].