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On $\boldsymbolψ$-amicable numbers and their generalizations

S. I. Dimitrov

TL;DR

This work studies the distribution of $\psi$-amicable numbers arising from Dedekind's $\psi$-function, proving that the set of such pairs has zero asymptotic density by adapting Erdős’ method to the multiplicative $\psi$-function. It develops a general framework for $\psi$-amicable $k$-tuples, including a constructive criterion that yields $aN_1,\dots,aN_k$ as a $\psi$-amicable tuple when coprimality and ratio conditions hold. Additionally, it presents an alternative definition of $\psi$-amicable $k$-tuples and catalogs extensive explicit examples for triples, quadruples, quintuples, and sextuples, linking to relevant OEIS sequences. Overall, the paper extends amicable-type phenomena from the divisor-sum function to Dedekind’s $\psi$, demonstrating sparsity while providing systematic construction approaches and rich tabulated instances.

Abstract

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

On $\boldsymbolψ$-amicable numbers and their generalizations

TL;DR

This work studies the distribution of -amicable numbers arising from Dedekind's -function, proving that the set of such pairs has zero asymptotic density by adapting Erdős’ method to the multiplicative -function. It develops a general framework for -amicable -tuples, including a constructive criterion that yields as a -amicable tuple when coprimality and ratio conditions hold. Additionally, it presents an alternative definition of -amicable -tuples and catalogs extensive explicit examples for triples, quadruples, quintuples, and sextuples, linking to relevant OEIS sequences. Overall, the paper extends amicable-type phenomena from the divisor-sum function to Dedekind’s , demonstrating sparsity while providing systematic construction approaches and rich tabulated instances.

Abstract

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

Paper Structure

This paper contains 6 sections, 7 theorems, 35 equations.

Table of Contents

  1. Notations, definitions and formulas
  2. Introduction and statement of the results
  3. Lemmas
  4. Proof of Theorem \ref{['Theorem']}
  5. $\psi$-amicable $k$-tuples
  6. Another definition of $\psi$-amicable $k$-tuples

Key Result

Theorem 1

Let $M(n)$ denote the number of $\psi$-amicable pairs $(a, b)$ with $a<b$ and $a\leq n$. Then $M(n)=o(n)$ as $n\rightarrow \infty$.

Theorems & Definitions (13)

  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • ...and 3 more