On $\boldsymbol{\psi}$-amicable numbers and their generalizations

TL;DR

This paper investigates the properties and distribution of $oldsymbol{\psi}$-amicable numbers, proving their asymptotic density is zero and proposing broader generalizations of these numbers.

Contribution

It introduces the concept of $oldsymbol{\psi}$-amicable numbers, proves their asymptotic density is zero, and presents new generalizations of these numbers.

Findings

Asymptotic density of $oldsymbol{\psi}$-amicable numbers is zero.

Proposed generalizations extend the concept of $oldsymbol{\psi}$-amicable numbers.

Theoretical analysis of properties of generalized $oldsymbol{\psi}$-amicable numbers.

Abstract

In this article, we study the properties of $\psi$-amicable numbers. We prove that their asymptotic density relative to the positive integers is zero. We also propose generalizations of $\psi$-amicable numbers.