On perfect, amicable, and sociable chains

TL;DR

This paper classifies all perfect, amicable, and sociable chains based on a specific operator that counts the occurrences of each number within the chain, extending classical number theory concepts to combinatorial sequences.

Contribution

It provides an exhaustive classification of all such chains, extending the concepts of perfect, amicable, and sociable numbers to chains.

Findings

List of all perfect chains identified

List of all amicable chains identified

List of all sociable chains identified

Abstract

Let $x = (x_0,...,x_{n-1})$ be an n-chain, i.e., an n-tuple of non-negative integers $< n$. Consider the operator $s: x \mapsto x' = (x'_0,...,x'_{n-1})$, where x'_j represents the number of $j$'s appearing among the components of x. An n-chain x is said to be perfect if $s(x) = x$. For example, (2,1,2,0,0) is a perfect 5-chain. Analogously to the theory of perfect, amicable, and sociable numbers, one can define from the operator s the concepts of amicable pair and sociable group of chains. In this paper we give an exhaustive list of all the perfect, amicable, and sociable chains.