The aliquot constant
type: Item
instance of: scholarly article
zbMATH Open document ID: 1259.11088
author: Wieb Bosma
author: Ben Kane
published in: The Quarterly Journal of Mathematics
publication date: 2012-06-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/0912.3660
full work available at URL: http://hdl.handle.net/10722/192200
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11Y70
Mathematics Subject Classification ID: 11A25
zbMATH DE Number: 6045445
zbMATH Keywords: aliquot divisor
zbMATH Keywords: aliquot sequence
zbMATH Keywords: sum of divisors
zbMATH Keywords: growth factor
zbMATH Keywords: geometric mean
zbMATH Keywords: Catalan-Dickson conjecture
describes a project that uses: Magma
MaRDI profile type: Publication
OpenAlex ID: W1976197351
arXiv ID: 0912.3660
DOI: 10.1093/QMATH/HAQ050
title: The aliquot constant
review text: Let \(s(n)\) denote the sum of the proper divisors of a natural number \(n\): NEWLINE\[NEWLINE s(n)=\sum_{{d|n}\atop{1\leq d<n}}d. NEWLINE\]NEWLINENEWLINENEWLINEThe sequence of numbers \( n,s(n),s^2(n),s^3(n),\dots \) is called an aliquot sequence with a starting value \(n\).NEWLINENEWLINEIn the paper, it is shown that the limit NEWLINE\[NEWLINE \lim_{N\rightarrow\infty}\frac1{N}\sum_{n=1}^N\log\frac{s(2n)}{2n}=\lambda NEWLINE\]NEWLINE exists and that this limit does not exceed \(-0.03\). It follows from this that, the geometric mean of \(s(2n)/2n\), the growth factor of the function \(s\) on even numbers, in the long run tends to be less than \(1\). In some sense, this fact may be taken as probabilistic evidence that aliquot sequences with an even starting value tend to be bounded.
reviewed by: Jonas Šiaulys
description: scientific article; zbMATH DE number 6045445