A Generalization of the Practical Numbers

Abstract

A positive integer n is practical if every m <= n a 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 certain 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.

Publisher

World Scientific Publishing

Publication Date

6-1-2018

Publication Title

International Journal of Number Theory

Department

Mathematics

Document Type

Article

DOI

https://dx.doi.org/10.1142/S1793042118500902

Keywords

Practical number, Arithmetic function, Panarithmic number, Sum-of-proper-divisors function

Language

English

Format

text

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