On the distribution of sociable numbers
type: Item
instance of: scholarly article
title: On the distribution of sociable numbers
zbMATH Open document ID: 1241.11120
author: Mitsuo Kobayashi
author: Paul Pollack
author: Carl Pomerance
published in: Journal of Number Theory
publication date: 2009-06-23T00:00:00Z
review text: Let \(s_1(n)= \sigma(n)- n\), the sum of the proper divisors of \(n\), and define \(s_j(n)= s(s_{j-1}(n))\) for \(j\geq 2\) and \(s_{j-1}(n)> 0\). If \(s_k(n)= n\) for some \(k\), then \(n\) is a sociable number and the least such \(k> 0\) is its order. The sociable numbers of order 1 are the perfect numbers and those with order 2 give amicable pairs, the existence of both of which was known in antiquity. The aim of this paper is to make progress towards the conjecture that the set of sociable numbers has asymptotic density zero. The authors establish some ten theorems, in most of which they estimate the number of sociable numbers \(n\leq x\) with a specific property; thus they show that this number is \(o(x)\) when one of the following conditions holds: (1) For fixed \(k\), including \(k= 0\), at most \(k\) numbers \(s_j(n)\) exceed \(x\); (2) The order of \(n\) is at most \(k\); (3) \(n\) is deficient, i.e. \(s(n)< n\); (4) \(n\) is even and abundant, i.e. \(s(n)> n\). It follows that the set of odd abundant numbers, the density of which is shown to be approximately \({1\over 500}\), contains all sociable numbers except for a set of density 0. The relevant work of many mathematicians, including that of Erdős, is described in the introduction with key papers included in the long and useful list of references.
reviewed by: Eira J. Scourfield
Mathematics Subject Classification ID: 11N56
Mathematics Subject Classification ID: 11A25
zbMATH DE Number: 5569037
zbMATH Keywords: sociable numbers
zbMATH Keywords: aliquot sequences
zbMATH Keywords: sociable \(k\)-cycle
zbMATH Keywords: perfect numbers
zbMATH Keywords: amicable numbers
MaRDI profile type: Publication
full work available at URL: j.jnt.2008.10.011
OpenAlex ID: W1999947597
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DOI: 10.1016/J.JNT.2008.10.011
description: scientific article; zbMATH DE number 5569037