Numerical and Statistical Analysis of Aliquot Sequences

Abstract

We present a variety of numerical data related to the growth of terms in aliquot sequences, iterations of the function s(n) = σ(n) − n. First, we compute the geometric mean of the ratio s k (n)/s k−1 (n) of kth iterates for n ≤ 2 37 and k = 1, . . . , 10. Second, we extend the computation of numbers not in the range of s(n) (called untouchable) by Pollack and Pomerance [2016] 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 − 1st iterates of s(n) but not kth iterates) along with some numerical data. Finally, inspired by earlier work of Devitt [1976], we estimate the growth rate of terms in aliquot sequences using a Markov chain model based on data extracted from thousands of sequences.

The Man Who Loved Problems: Richard K. Guy

“…This was more or less predicted by Guy's calculations and heuristic models using Markov chains. The theory continues for 𝑠(𝑠(2𝑛))/𝑠(2𝑛), but beyond the second iterate we only have these heuristics, [7]. Richard intrigued bright young minds into working on this problem for over 40 years, and at one point described it as his favorite problem.…”

“…Guy proved several conjectures on crossing numbers for different families of graphs. He also 7 From an essay of Ted Bisztriczky in a collection of Guy remembrances in the CMS Notes, September, 2020. looked at the slimming number, the minimal number of edges one needs to remove to make 𝐺 planar.…”

Richard Kenneth Guy, 1916–2020

“…From a combination of theory and numerical evidence, Guy and Selfridge made a counter‐conjecture, that $$\lbrace s^k(n)\rbrace _{k=1}^\infty$$ diverges for many, if not all, even $$n$$. Richard's last paper [129] co‐authored with some computer scientists, provided much stronger numerical evidence of this, including investigating 8000 sequences with randomly selected initial numbers until they terminated or exceeded $$2^{888}$$.…”

“…Catalan in 1888, with a modification by Dickson in 1913, had conjectured that all such sequences terminate, that is reach a prime 𝑝, so that 𝑠(𝑝) = 1, or become periodic. From a combination of theory and numerical evidence, Guy and Selfridge made a counter-conjecture, that {𝑠 𝑘 (𝑛)} ∞ 𝑘=1 diverges for many, if not all, even 𝑛. Richard's last paper [129] co-authored with some computer scientists, provided much stronger numerical evidence of this, including investigating 8000 sequences with randomly selected initial numbers until they terminated or exceeded 2 888 .…”

An Early-Warning Model for Online Learners Based on User Portrait

“…In terms of data analysis technology, the behavior features of online learners are usually mined from multi-dimensional data on their participation, interaction, and psychological features, through statistical analysis [7], sequence analysis [8], association rule mining [9], social network analysis [10]. Considering the features of online learning, Wang et al [11] proposed an ant colony optimization (ACO) algorithm, in which the pheromone concentration is adjusted step by step, to optimize the recommendation of learning path and predict the learning performance in future.…”