20211029, 09:49  #1 
Nov 2011
271 Posts 
Numerical and Statistical Analysis of Aliquot Sequences
I want to share a paper, where the authors did a great job in investigating a behaviour of aliquot sequences. Well, it was published in Experimental Mathematics in 2020, but only appeared on arxiv several days ago.
The main objective of the paper is to find an evidence in favour of one of the following hypotheses:
The computations show that the geometric mean of where n is even and changes from 1 to N grows with k but decays with N. For example, for the values of are smaller than 1 for k up to 5 and bigger than 1 for bigger values of k. Then the authors tried some smarter ways of computing the geometric mean. For example, they add weightings to terms according to the number of aliquot preimages of n. Hence, untouchable numbers are not counted in the mean but if a number has ten preimages it is counted ten times. After such weightings are implemented, the geometric mean becomes even smaller. The conclusion of this part of the paper is that most probably the values and their weighted analogues become smaller than 1 as N becomes large enough. That is a good sign for everyone who wants to terminate all aliquot sequences. But in the second part of the paper, the authors take into account the guides and drivers, i.e. the fact that aliquot sequences tend to stick with certain factorisation patterns and are very hard to escape from them. The authors considered 8000 aliquot sequences and extended them until they terminate or become larger than . The data coming from those sequences indicate that the downdriver is the most frequent driver, but the number of all the other (up)drivers is way bigger than that of downdrivers. And also the geometric mean value of for the terms of those sequences is around 1.133. That is a rather strong evidence towards Conjecture (b). An interesting observation from the paper's data: the downdriver appears to be the easiest driver to go away from. The average length of the downdriver is 30.488. It is followed by with the average length 33.139, then , , , and the stickiest driver is with the average length 188.307. 
20211030, 17:09  #2 
"Daniel Jackson"
May 2011
14285714285714285714
2A8_{16} Posts 
Which sequences did they run? Any chance of getting a txt file with all the sequence starters in it?

20211030, 21:21  #3  
"Ed Hall"
Dec 2009
Adirondack Mtns
47·89 Posts 
Quote:


20211031, 02:26  #4  
Jun 2003
3^{4}×5×13 Posts 
Quote:


20211031, 04:00  #5  
Nov 2011
10F_{16} Posts 
Quote:
each sequence starting at , where 0 ≤ n ≤ 7 and 0 ≤ k < 1000." I would suggest to try to contact the authors and ask them for data. Last fiddled with by Drdmitry on 20211031 at 04:53 

20211031, 13:51  #6  
"Ed Hall"
Dec 2009
Adirondack Mtns
10127_{8} Posts 
Quote:
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Code:
as evidence that most even aliquot sequences diverge But, it's obviouslyly just me. . . 

20211031, 17:56  #7  
"Garambois JeanLuc"
Oct 2011
France
2^{6}×11 Posts 
Quote:
"all odd terminate" This is not proven ! I would not dare to formulate the conjecture that there are sequences that have only odd terms and are OpenEnd. But I would gladly formulate (unofficially of course) the following conjecture : There exist sequences with all odd terms that are increasing for k iterations with k as large as one wants. I looked for such sequences using the "regina" file and the record sequence I found is the one starting with 2551185 and whose terms are all odd and which is increasing until index 6. And this starting odd number has only 7 digits. I can easily assume that other odd numbers of millions of digits could be the start of increasing oddterm sequences for thousands or millions of iterations. For the even sequences, I side with Guy and Selfridge. A few years ago, with some friends, we wrote two pdf documents and even today, I have three computer programs constantly running to update these pdf documents. But it will take a few years before the calculations are accurate enough. And these works make me intuitively lean towards the existence of divergent sequences. http://www.aliquotes.com/vitesse_croissance.pdf http://www.aliquotes.com/infirmer_catalan.pdf Thank you very much in any case for letting us know about this new article, I will read it very carefully. 

20211031, 20:07  #8 
"Ed Hall"
Dec 2009
Adirondack Mtns
4183_{10} Posts 

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