A Generalization of the Practical Numbers

A positive integer n is practical if every m \u3c= n a can be written as a sum of distinct divisors of n. One can generalize the concept of practical numbers by applying an arithmetic function f to each of the divisors of n and asking whether all integers in a certain interval can be expressed as sums of f(d)\u27s, where the d\u27s are distinct divisors of n. We will refer to such n as f-practical . In this paper, we introduce the f-practical numbers for the first time. We give criteria for when all f-practical numbers can be constructed via a simple necessary-and-sufficient condition, demonstrate that it is possible to construct f-practical sets with any asymptotic density, and prove a series of results related to the distribution of f-practical numbers for many well-known arithmetic functions f