On practical sets and $A$-practical numbers

TL;DR

This paper investigates the properties and structure of $A$-practical numbers, a generalization of practical numbers based on a subset $A$ of positive integers, and explores the behavior of the associated set mapping.

Contribution

It introduces the concept of $A$-practical numbers, analyzes their properties, and studies the set-theoretic and dynamic aspects of the mapping from subsets of natural numbers to $A$-practical numbers.

Findings

Characterization of the form and size of $ ext{Pr}(A)$ for various $A$

Analysis of the set-theoretic properties of $ ext{Pr}(A)$

Investigation of the dynamics of the mapping $ ext{Pr}: ext{PowerSet}( ) o ext{PowerSet}( )"

Abstract

Let $A$ be a set of positive integers. We define a positive integer $n$ as an $A$-practical number if every positive integer from the set $\left\{1,\ldots ,\sum_{d\in A, d\mid n}d\right\}$ can be written as a sum of distinct divisors of $n$ that belong to $A$. Denote the set of $A$-practical numbers as $\text{Pr}(A)$. The aim of the paper is to explore the properties of the sets $\text{Pr}(A)$ (the form of the elements, cardinality) as $A$ varies over the power set of $\mathbb{N}$. We are also interested in the set-theoretic and dynamic properties of the mapping $\mathcal{PR}:\mathcal{P}(\mathbb{N})\ni A\mapsto\text{Pr}(A)\in\mathcal{P}(\mathbb{N})$.