# A generalization of the practical numbers

## TL;DR

This paper introduces the concept of $f$-practical numbers, generalizing practical numbers by applying an arithmetic function to divisors, and explores their construction, density, and distribution for various functions.

## Contribution

It defines $f$-practical numbers, provides criteria for their construction, and analyzes their distribution and density for different arithmetic functions.

## Key findings

- Criteria for constructing $f$-practical numbers
- Existence of $f$-practical sets with any asymptotic density
- Distribution results for $f$-practical numbers

## Abstract

A positive integer $n$ is practical if every $m \leq n$ 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 given interval can be expressed as sums of $f(d)$'s, where the $d$'s 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 $f$.