On practical sets and A-practical numbers

2024/05/28 by Andrzej Kukla, Kukla, Andrzej, Piotr Miska +1
Computer Science · #AI-based Problem Solving and Planning #Constraint Satisfaction and Optimization #FOS: Mathematics #Number Theory (math.NT) #Rough Sets and Fuzzy Logic

paper · pdf · doi:10.48550/arxiv.2405.18225

openalex publication_date 2024/05/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \1,… ,∑_{d∈ A, d| 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 Pr(A). The aim of the paper is to explore the properties of the sets Pr(A) (the form of the elements, cardinality) as A varies over the power set of ℕ. We are also interested in the set-theoretic and dynamic properties of the mapping PR:P(ℕ)\ni A\mapstoPr(A)\inP(ℕ).

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