Aliquot sequences

Welcome to Juan L. Varona WWW page about aliquot sequences.

First of all, sorry for the English in this page; I am sure it will have some spelling and grammar mistakes. Also, note that the notation for some mathematical formulas follows the one of TeX.

For a positive integer n, let sigma(n) denote the sum of the divisors of n, and s(n) = sigma(n) - n. A perfect number is a number n such that s(n) = n, and an amicable pair of numbers is (n,m) satisfying s(n) = m, s(m) = n. In a similar way, cycles of numbers (a_1, a_2, ... , a_p) such that s(a_i) = a_{i+1} for i = 1, ... , p-1 and s(a_p) = a_1 are known as aliquot cycles or sociable numbers.

Iterating with the function s, it appears a aliqout sequence. For instance, starting from n = 20, s(20) = 1+2+4+5+10 = 22, s(22) = 1+2+11 = 14, s(14) = 1+2+7 = 10, s(10) = ... and so on. For one of such sequences, there are four possibilities:

(i) it terminates at 1 (the previous term being a prime),
(ii) it reaches a perfect number,
(iii) it reaches an amicable pair or a cycle,
(iv) it is unbounded.

Catalan-Dickson conjecture says that (iv) does not actually happen. But other researchers disagree with this conjecture and they think that there are unbounded sequences; in fact, the alternative conjecture from Guy-Selfridge states that there are a lot of sequences that go to infinity, perhaps almost all those that start at an even number.

The smallest n for which there was ever doubt was 138, but Lehmer showed that the sequence terminates at s^{177}(138) = 1. Since then, the first number whose behavior is not known is 276. Many researchers have investigated the behavior of this sequence, but it is not yet known if its end exists.

But there are many other sequences whose end is in doubt. For instance, there are five main sequences starting in a number smaller than 1000 whose behavior is unknown (Lehmer five: 276, 552, 564, 660 and 966). Before the year 1980, there were fourteen main sequences starting between 1000 and 2000 (Godwin fourteen), but at the present time only twelve remain unknown (Godwin showed that the 1848 sequence terminates, and so did Dickerman for the 1248 one).

In 1994, A. W. P. Guy and R. K. Guy published a article with a table showing the status of the sequences starting with numbers less than or equal to 7044. In the same year, the book Unsolved problems in Number Theory (2nd ed.), from R. K. Guy, updates the information of the article. In this book, a extensive bibliography on this subject can be found.

Since then, Manuel Benito and myself have been checking the aliquot sequences for numbers under 10000. For some starting values, we have shown for the first time that the sequence terminates. In December 22, 1996, we found the record for the maximum of a terminating sequence: the one starting at 4170 converges to 1 after 869 iterations getting a maximum of 84 decimal digits at iteration 289.

In Octuber 1999, Wieb Bosma beated this record: he found that the aliquot sequence starting with 44922 terminates after 1689 iterations (at 1) after reaching a maximum of 85 digits at step 1167. Later, in December 3, 1999, he beated again the record finding that the sequence starting at 43230 finished: it terminates (at 1) after 4357 steps after reaching a maximum of 91 digits at step 967.

On June 10, 2001, Manuel Benito and myself got again the record for the highest known terminating sequence. The sequence starting at 3630 reaches a maximum of 100 digits at index 1263, and ends (at 1) at step 2624 (with the prime 59 being the previous term). Later, in December 25, 2001, we found that the sequence starting at 6160 finished: it terminates (at 1) after 3027 steps (with the prime 601 being the previous term) after reaching a maximum of 96 digits at step 1631.

In our web pages, we will deal only with sequences starting under 10000. At present time, the aliquot sequences starting in a number under 10000, and whose end is yet unknown are the following: 276, 552, 564, 660, 966, 1074, 1134, 1464, 1476, 1488, 1512, 1560, 1578, 1632, 1734, 1920, 1992, 2232, 2340, 2360, 2484, 2514, 2664, 2712, 2982, 3270, 3366, 3408, 3432, 3564, 3678, 3774, 3876, 3906, 4116, 4224, 4290, 4350, 4380, 4788, 4800, 4842, 5148, 5208, 5250, 5352, 5400, 5448, 5736, 5748, 5778, 6396, 6552, 6680, 6822, 6832, 6984, 7044, 7392, 7560, 7890, 7920, 8040, 8154, 8184, 8288, 8352, 8760, 8844, 8904, 9120, 9282, 9336, 9378, 9436, 9462, 9480, 9588, 9684, 9708, 9852. These sequences have have pursued to at least 100 digits.

To get more information about these sequences:

Table that summarize the present state of these sequences.
Occurrences of numbers of 80 digits, 90 digits, and 100 digits in these sequences.
The sequences themselves in the ftp server.

All our work has been done by using free packages available on internet. We have run the programs on many computers from the authors and some colleages, and their respective institutions.

We have used the following packages, that are available at their corresponding web pages (or anonymous ftp sites):

If you are interested in this subject, you can found pre-print with our results (in tex, dvi and pdf formats), the last term for any doubtful sequence, and the present state of the calculus by using anonymous ftp from mat.unirioja.es/pub/aliquot.

Here, you can download a simple PARI-GP program or KASH program that allow to compute an aliquot sequence.

In any case, feel free to contact us; perhaps the results in these web pages and/or the ftp server are not up to date.

Other people that is involved in this subject:

Wolfgang Creyaufmueller, from Aachen, is cumputing aliquot sequences starting a in number between 10^5 and 10^6 up to a minimum of 60 digits. This work is documented in his website, both in English and in German.
Paul Zimmermann is computing iterates of the sequences starting in the numbers 276, 552, 564, 660, 966, 1074, 1134 and 204828. Also, he is working with aliquot sequences starting between 50000 and 100000. He maintains a web page.
Jim Howell is working in the aliquot sequence starting in 966. The status can be seen in his aliquot page
Wieb Bosma is working with aliquot sequences starting between 10000 and 50000. He also maintains a web page.
Christophe Clavier is continuing with sequences starting in a number under 10000. See his web page.
Mitchell Dickerman (who completed the 1248 sequence) is working with unitary aliquot sequences (one takes only the unitary divisors, i.e. those divisors d which do not divide n/d).

Last modification of this page: September 16, 2004.

jvarona@dmc.unirioja.es

Personal home page (in Spanish): http://www.unirioja.es/cu/jvarona/hola.html

Personal home page (in English): http://www.unirioja.es/cu/jvarona/welcome.html