On practical sets and $A$-practical numbers

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 $\mathcal{PR}:\mathcal{P}(\mathbb{N})\ni A\mapsto\text{Pr}(A)\in\mathcal{P}(\mathbb{N})$.