On practical sets and $A$-practical numbers

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 $\{1,\dots ,{\sum }_{d\in A,d\mid n}d\}$ 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 $\mathcal{P}\mathcal{R}:\mathcal{P}(\mathbb{N})\ni A\mapsto \text{Pr}(A)\in \mathcal{P}(\mathbb{N})$.