Numerical and Statistical Analysis of Aliquot Sequences

We present a variety of numerical data related to the growth of terms in aliquot sequences, iterations of the function \(s(n) = \sigma(n) - n\). First, we compute the geometric mean of the ratio \(s_k(n)/s_{k-1}(n)\) of \(k\)th iterates for \(n \leq 2^{37}\) and \(k=1,\dots,10.\) Second, we extend t...

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Bibliographic Details
Published in:arXiv.org 2021-10
Main Authors: Chum, Kevin, Guy, Richard K, Jacobson, Michael J, Jr, Mosunov, Anton S
Format: Article
Language:English
Subjects:
Algorithms
Markov chains
Numbers
Sequences
Statistical analysis
Online Access:Get full text
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Summary:We present a variety of numerical data related to the growth of terms in aliquot sequences, iterations of the function \(s(n) = \sigma(n) - n\). First, we compute the geometric mean of the ratio \(s_k(n)/s_{k-1}(n)\) of \(k\)th iterates for \(n \leq 2^{37}\) and \(k=1,\dots,10.\) Second, we extend the computation of numbers not in the range of \(s(n)\) (called untouchable) by Pollack and Pomerance to the bound of \(2^{40}\) and use these data to compute the geometric mean of the ratio of consecutive terms limited to terms in the range of \(s(n).\) Third, we give an algorithm to compute \(k\)-untouchable numbers (\(k-1\)st iterates of \(s(n)\) but not \(k\)th iterates) along with some numerical data. Finally, inspired by earlier work of Devitt, we estimate the growth rate of terms in aliquot sequences using a Markov chain model based on data extracted from thousands of sequences.
ISSN:2331-8422
DOI:10.48550/arxiv.2110.14136