Numerical and Statistical Analysis of Aliquot Sequences

TL;DR

This paper provides extensive numerical analysis of aliquot sequences, including growth patterns, untouchable numbers, and a Markov chain model to estimate their growth rates, advancing understanding of their long-term behavior.

Contribution

It introduces new computational methods and data for analyzing aliquot sequences, including algorithms for untouchable numbers and a Markov chain model for growth estimation.

Findings

Computed geometric means of ratios for aliquot sequence iterates up to 2^37.

Extended the range of untouchable numbers to 2^40 and analyzed their properties.

Developed a Markov chain model to estimate the growth rate of aliquot sequence terms.

Abstract

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.