Aliquot Sequences

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Definition - Catalan Conjecture

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An aliquot sequence is a sequence of integers, built with the sigma function. sigma(n) or s(n) is the sum of divisors of an integer n.
The sum
of the proper divisors is 

i(n) = sigma(n) - n

Iterate: i(n) = sigma(n) - n, i(i(n)) = sigma(i(n)) - i(n) and so on. 276:i4 means the fourth iteration in the sequence starting with 276. This sequence begins with 276, 396, 696, 1104, 1872 - 276:i4 is 1872. You can look at the complete sequence with this link: 276 or at the graph.

Normally an aliquot sequence ends in a prime. Different sequences can come together and end in the same prime. All these side sequences are called a prime family (Primzahlfamilie). New calculations occasionally lead to a confluence of two former different aliquot sequences into one family.

This was first published by the Belgian mathematician Eugène Catalan in the year 1888. Leonard Eugene Dickson extended the so called Catalan conjecture: "Each aliquot sequence ends in a prime, in a perfect number or in an aliquot cycle".

Up to now it is not possible to certify the Catalan conjecture.

Each confluence of two sequences gives some more hope, but it's no proof of the conjecture, it's only some work on the way to possible solution. A great step in this direction was done in February 2005 from Christophe Clavier. He found the confluence of sequence 1578 and 56440 with a record descent from C110 down to C5

The term "Catalan's Conjecture" is used for another mathematical problem, too: 
Catalan's conjecture states that the equation  xm - yn = 1  has no other integer solution but 32 - 23 = 1. In May 2002 Preda Mihailescu gave a proof for this conjecture. On this website you can't find anything else about these facts.

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Aliquot Sequences

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An ending aliquot sequence is a so called terminating sequence. Normally the end of an aliquot sequence is a prime.

Instead of a prime an aliquot sequence can end in a cycle, too. Well known are amicable pairs. Pythagoras said true friendship is comparable to the numbers 220 and 284 - this is the smallest amicable pair: i(220) = 284 and i(284) = 220. Meanwhile thousands of amicable pairs are found. Most of them are constructed with the help of Thabit rules.
Pedersen
counted more than 11994387 (on October, 1st, 2007) amicable pairs mostly in the interval [1, 10^999],  some of them in [10^1000, 10^13581]. Up to 10^14 the listing is exhaustive. You will find links to all pair and notes to the explores.
Note: Pedersen's data get updated in short intervals.

Some cycles are of higher order (so called sociable numbers).  Cycles with 4, 5, 6, 8, 9 and 28 members are known. Other orders are possible, too. Today (April 2013) we know 221 aliquot cycles with higher order. 
David Moews
shows a complete listing of all known aliquot cycles,  Jan Pedersen shows a comparable table, too. The sequence with the key-number 17490 ends in an aliquot-4-cycle. There are 206 cycles of the order 4, five of the order 6, three of the order 8, one of the order 5, 9 and 28. Up to 5*10^12 the listing for aliquot-4-cycles is exhaustive. The largest 4-cycle has 71 digits.

The smallest cycles are the perfect numbers. Their sum of divisors is doubled n and i(n) = n. The smallest perfect numbers are 6, 28, 496, 8128, 33550336. Today there are 48 perfect numbers known (Mersenne prime no. 42 was found in February 2005, no. 43 in December 2005,  no. 44 in September 2006, no.45 in August 2008, no. 46 in September 2008, no. 47 in June 2009, no. 48 in January 2013). Their greater prime factor is a Mersenne prime. To my knowledge it is not yet known whether all perfect numbers are even or not. Brent, Cohen and te Riele gave a lower bound of 10^300 for odd perfect numbers.
There is a search of such numbers (
GIMPS) in the world wide web. The Mersenne prime no. 43 has 9152052 digits, no. 44 has 9808358 digits, no. 45 has 12978189 digits. The others are a little bit smaller (for details have a look on GIMP, please). The record is no. 48 with 17425170 digits.

Several sequences are not computed up to their end. They are increasing and no one knows if they will end or not. The smallest start-up number (or key-number, beginning number) of such a so-called open-end sequence (OE-sequence - Offenendkette) is
276 (-> labyrinth of Chartres). An open-end sequence with all side sequences is called an open-end family (OE-family).

There are 5 open-end sequences in the interval [1, 1000] with the key numbers 276, 552, 564, 660 und 966. They are called Lehmer FiveThere are 81 open-end-sequences in the interval [1, 10^4].
There are now 898 open-end sequences in [1, 10^5] and  9202 OE-sequences in [1, 10^6].
Any progress in calculation can reduce these numbers.

The following table shows the actual limits of computation (B = Bosma, C = Creyaufmueller, CL = Clavier, G = Gerved, H=Hoogendoorn, S = Stern, VB = Varona/Benito, Z = Zimmermann) and the number of open-end sequences. A detailed table shows more intervals. Varona gives an overview at [1,10^4]. New are the tables for [1, 220000].

Table - overview :

interval

number of OE- sequences

limits of computation

computed by

       
[1, 1000]       5 > 10^157 C/Z
[1, 10000]     81 > 10^150 C/VB/Z/CL
[1, 50000]   440 > 10^100 B/C/G/VB
(50000, 10^5]

    458

> 10^100 C/G/S/Z
[1, 100000]   898 > 10^100 B/Z/G/VB/S/C/CL
(100000, 200000]   950 > 10^100 C/VB
(200000, 300000]   911 > 10^80 / 10^100 C / H / B
(300000, 400000]   851 > 10^80 / 10^100 C/B
(400000, 500000]   883 > 10^80 C
(500000, 600000]   945 > 10^80 C
(600000, 700000]   936 > 10^80 C
(700000, 800000]   920 > 10^80 / 10^100 C
(800000, 900000]

  954

> 10^80 C
(900000, 10^6]   954 > 10^80 C
       
[1, 10^6] 9202  > 10^80 / 10^100   
Download all OE-Sequences > 10^80 / 10^100  

A group of free workers will do lots of work on aliquot sequences. To avoid multiple work please check the website of Mersenne-Forum for reserved sequences! (200k, 250k] is up to >C100. (500k, 600k] is reserved.

detailed table [1, 10^6]  

About 1% of all integers are beginning numbers (key numbers) of an open-end sequence. This is an empirical result. Have a look at the complete statistic in [1, 10^6], click here for download.
Attention: Please have a look on Richard K. Guy's 'Law of Small Numbers'.

The table above is the result of long calculations. The program Aliquot by Ivo Duentsch produces a record file (so-called matrix). Each integer is related to its target. Each aliquot sequence was computed up to its end or up to a break, if the sequence grows too much. These open-end sequences were computed separately with UBASIC-programs. The data for the interval [1, 10^6] have increased up to several Gigabytes during several years.
All sequences were checked for combinations. Side sequences got the newly found prime. Side sequences of OE-sequences got the smallest key-number comparable to a target-prime.
Each sequence in the whole interval up to 10^6 was checked up to 40 digits in a first step. In a second step this upper limit was expanded to a minimum of 60 digits. These complex calculations were finished in May 1999. The third step is the expanded calculation up to 80 digits. These calculations were finished end of March 2003.
In the meantime a few more sequences have been terminated. The table above includes all known terminating sequences in the statistic.
The extension of calculation limit from C60 up to C80 reduces the number of OE-sequences about 2 - 2.5%. Some sequences terminate, the other were identified as side-sequences.
At Christmas 2001 Varona terminated the sequence 6160 and 1797 numbers in (1, 10^6] changed  their family status.

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Diagrams

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To get a general idea a graphic presentation of an aliquot sequence is helpful using a semi logarithm axis, i.e. a linear x-axis beginning with 0 for the index number of the sequence and the y-axis on a scale of decadic logarithm for the sum of the proper divisors. 
So there is a function f: N(n) -> log10 i(n).
There are three types of aliquot sequences and three types of diagrams, too.

 

1) terminating sequences The graph is a single irregular peak (ending in a prime number) Example: g840
2) cyclic ending The graph ends in a horizontal line (ending in a perfect number) Example: g976950
  The graph alternates  between two horizontal asymptotic lines (ending in a pair of amicable numbers) Example: g980460
  The graph alternates  between two horizontal asymptotic lines (ending in an aliquot cycle) Example: g17490 und g2856
3) open-end sequence The graph ascends and has it's maximum often at the last calculated term Example: g276

Additional diagrams will be seen with the help of the links on the  record list or on the additional page "Lehmer Five".

up    840    976950    980460    17490    2856    276   1578    down 

840 - terminating sequence (complete sequence)

Graph 840

up    840    976950    980460    17490    2856    276   1578   down 

976950 - ending in perfect number (6)

Graph 976950

up    840    976950    980460    17490    2856    276   1578   down 

980460 - ending in an amicable pair (2620/2924)

Graph 980460

up    840    976950    980460    17490    2856    276   1578   down 

17490 - ending in an aliquot 4-cycle

Graph 17490

up    840    976950    980460    17490    2856    276   1578   down 

2856 - ending in the aliquot 28-cycle

Graph 2856

up    840    976950    980460    17490    2856    276   1578   down 

276 - open-end sequence

Graph 276

up    840    976950    980460    17490    2856    276   1578   down 

1578 - OE-sequence with deep minimum

Diagrams of Genealogy

(for greater picture click here or above)

up    840    976950    980460    17490    2856    276   1578   down 

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Databases

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1) The data of the aliquot sequences in [1, 1000000]  includes the data base C9C30. Here you'll find all information about terminating and open-end sequences. There you will find the first 9digit-number (C9) and the first 30digit-number (C30) of all sequences. With this data it will be possible to identify side-sequences.

2) A second data  base C60 shows the first 60digit-number of every sequence.  It will be possible to identify some more side-sequences.

3) In spring 2000 calculations up to C80 began. A third database is in progress. The main part  was finished in march 2003. You will find the first part here: 
C80

C9C30 - part / C60 - part / C80 - part

C9C30 / C60 / C80 - complete (zipped for download)

4) Aliquot needs a matrix (record file). Each number corresponds to a target. There are different possibilities for an entry:
a) -1 if n has not yet been looked at
b) a prime if the sequence starting with n ends in this prime
c) a positive nonprime integer (the smallest member of an aliquot cycle) if the sequence ends in a perfect number, in an amicable pair or cycle
d) a negative number, if the sequence is an OE-sequence (the entry is the negative key-number)
With this categories you can sort all entries easily.

There are only entries of such sequences which were computed with UBASIC-programs in the base-matrix. It's easy to change entries in such a matrix. Its easy to calculate a base-matrix, too. To change a totally calculated record file is more difficult because not all side-sequences can be changed.

ALIQUOT and corresponding record files

 

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Internet - links

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The following internet pages give more information about this subject:

aliquot sequences - Primzahlfamilien

amicable numbers - aliquot cycles

Mersenne primes - perfect numbers

For German users the following websites are useful: A guide to the complex website of Chris Caldwell is the mask of Tobias Jentschke (in German) with effectively set links. The subject has been reviewed in a  newspaper article in Mario Jeckle's website. Both websites link directly to a page with a biography of Marin Mersenne. Udo Hebisch shows an actual table of Mersenne primes. Caldwell renders short biographies of living mathematicians on a further page. 

definition    aliquot sequences      tables    diagrams    databases    links    literature    factorizing    programs    projects    records    start    ©


Literature

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You will not find many books in German. There are only a few older mathematical articles. The following link will show you a bibliography of printed documents. 
David Moews,  Eric W. Weisstein and Chris K. Caldwell give an actual overview about the English literature. In German the following book summarizes most of all:

Wolfgang Creyaufmüller: "Primzahlfamilien - Das Catalan'sche Problem und die Familien der Primzahlen im Bereich von 1 bis 3000 im Detail", 19951, 176 p. / 19972, 262 p. / 20003, 327 p.
ISBN 3-9801032-2-6.

Here you will find a great bibliography and commented programs for PCs with complete source code. The following bibliography-link shows the actual mathematical part of the 3rd edition.

Verlagsbuchhandlung Creyaufmüller

An interesting website you will find by Scott Contini's FactorWorld. There are download-possibilities for many papers.


Factorizing great Integers

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Computing aliquot sequences creates problems in factorizing at an early stage. You seldom see this inside a terminating sequence with a low maximum. If the sum of divisors like in OE-sequences normally increases you will often find only smaller primes fast (example 276). There remains a greater rest - a composite integer - seldom with more than two prime factors. To factorize this composite number is hard and time-consuming. Normally you will not find any rule how to compute them, but you can crack the factors with the programs below.

ECM - Factorizing with elliptic curves

During the last few years a lot of factorizations have been carried out very effectively with ECM-method, a computing technique using elliptic curves. Programs on this base will find factors up to 20 digits fast. That means in minutes or in a few hours. Factors up to 30 digits you will normally find in a few hours, seldom in days. For greater factors you need luck! 
A new record is the 54digit prime found by ECM at 26. Dec. 1999. You will find the most interesting information on the
ECMNET page of Paul Zimmermann.

PPMPQS - Factorizing with multiple polynomial quadratic sieves

To factorize a really great integer you can use a multiple polynomial quadratic sieve (PPMPQS), too. For a 100digit-number you need a few weeks computation time. In autumn 2001 was the record at 109 digits. Use the links below for downloading the latest UBASIC versions of PPMPQS.

Number Field Sieve (Nfs)

Up to now I couldn't test the complete number field sieve myself. I couldn't integrate the number field sieve into the program for calculating aliquot sequences, too. But you will find literature and programs on the website of Conrad Curry (this link is dead from time to time) or on the website of Henrik Olsen
The best website in this days is Paul Leyland's. There you will find an excellent description of the whole theme.
Some papers you will find at nfs-Papers.
Good information you will find in mathworld, too.
An interesting site for a gnfs-implementation: GGNFS.

There is one method to run GGNFS in a DOS-Window under WinXP actually. NFS is god for C98 composites as minimum. The speed is pretty good, NFS runs about 20 times faster than ECM or PPMPQS. These values are approximately. 
A) First you have to install ActiveRearl:  Free Active Pearl
B) You need some binaries. You will find them under this link: Free GGNFS
C) You must install NFS with its folders.
D) All this you will find in this ZIP-file: ggnfs.zip
Start the number sieve in the subdirectory "TESTS" and execute num with the number you want to factorize.

There are sieves for special and sieves for arbitrary numbers (general nfs). In teamwork we cracked a C111-cofactor in the sequence 276 with Nfs, but each of us did only a part of the work with a part of the program. The record in May 2007 is a C307-factorization (C200-factorisation in May 2005), (C174-factorisation in December 2003), ( C158-factorisation in February 2003) by Jens Franke.
It exist a new implemented Ubasic version of Nfs from Yuji Kida. It runs only with the experimental version Ubasic 9.

Factorizing great integers is a part of cryptography and together with this a question of data security.

You will find a description of all common factorizing methods on the website of Jim Howell.

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Programs

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The programs I used are written in Turbo Pascal and in UBASIC.
UBASIC is especially for mathematical use and for computation with great numbers. It's normally freeware. You can factorize integers up to a limit of 105 digits with the DOS version of Yuji Kida's PPMPQS, a multiple polynomial sieve. The actual Windows version computes up to 120 digits . Both methods are integrated in my programs for aliquot sequences.

ALIQUOT and corresponding record files

UBASIC-programs (zipped for download) - last update: 20-4-2002/29-12-2003/12-6-2004

ALQ-Converter (zipped for download)

Other Programs

 

Comment:
ALIQUOT is written by Ivo Duentsch in Turbo Pascal and available in its original version at the ftp-server. The expanded German versions AQCN and AQSN for statistics are the base of the table above. Both programs were patched in February 2001. The old version created the well known "Borland-Turbo-Pascal-runtime-error 200".
The UBASIC-programs Ellixts.ub, Ellippmp.ub and Ellppmpx.ub are totally given in the source code. They are freeware, too. You can use them and test your own variations, for example at school ...
Ellippmp.ub and Ellppmpx.ub alternate automatically between calculation with Ecm and calculation with Ppmpqs.
Both converters Alq2elf und Elf2alq are written by Jesper Gerved. They are useful for condensing the data-files (*.ELF).
In December 2001 Clifford Stern wrote a helpful program to find out the confluence-point of side-sequences. A second program converts SQ-files into *.ELF-files.

Additional information

You will find other programs for other hardware in the web. Have a look on Richard Pinch's page for computer algebra. On my page are mainly information about DOS/Windows-computer and programs for this machines. Paul Zimmermann gives a lot of information about UNIX-computers and programs for calculations. On this page you will find many useful links, too. Some examples below:
Hisanori Mishima has a new factorization program from Satosi Tomabechi: Ppsiqs. It is said to be faster than Ppmpqs.

 ECM - PPMPQS - PPSIQS  NFS  General
 ECMNET - free available ECM programs  Number Field Sieve  Algorithmische Zahlentheorie
 ECM client/server  Number Field Sieve Org.  Jim Howell
 Hisanori Mishima  GGNFS  Richard Pinch's Computer Algebra Links
 Arithmétique Théorie des Nombres  Wikipedia-GNFS  MIRACL - Michael Scott
 ECC Tutorial  MSIEVE  Richard Brent's factor tables
 Elliptic Curves    Wilfried Keller - Faktortafeln
     Crypto-World
     Factorization Announcements
     numbertheory.org/ntw/N4.html - links
     Primzahlen - prime numbers
     Prime Links ++
     Primzahlseite

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School projects

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At school you can use this material in classes 9 up to 12 in mathematics for manual arithmetics. We've applied it several times with great success at Freie Waldorfschule Aachen. computer Sciences we teach in class 11 and 12. Pupils can make their own programs easily. The Project stopped in September 2008 due to energy prices.


Projects in Progress - History

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December 1999:
The programs for aliquot sequences (DOS version) have been running for several years without any trouble. The conversion to Windows (3.x, 9x) has been successful, too.

May/June 2000:
The Ubasic programs Ellixts.ub, Ellippmp.ub and Ellppmpx.ub have been modified for factorizations into three primes.
The calculations in (1134, 10000] (Varona) have been interrupted at indices between C91 and C100.
The calculations of 921232 (record aliquot sequence) have been interrupted at index 5326.

October 2000:
The programs Ellippmp.ub and Ellppmpx.ub got a bug fix.

December 2000:
All sequences in [900000, 10^6] are calculated up to a new limit: Generally C80. There are now 987 OE-sequences in this interval.

May 2001:

All sequences in [800000, 900000] are calculated up to a new limit: Generally C80. There are now 982 OE-sequences in this interval.

October 2001:
All sequences in [100000, 200000] are calculated up to a new limit: Generally C80. There are now 975 OE-sequences in this interval.

January 2002:
All sequences in [200000, 300000] are calculated up to a new limit: Generally C80. There are now 938 OE-sequences in this interval.

March 2002:

All sequences in [300000, 400000] are calculated up to a new limit: Generally C80. There are now 877 OE-sequences in this interval.

May 2002:
All sequences in [400000, 500000] are calculated up to a new limit: Generally C80. There are now 917 OE-sequences in this interval.

July 2002:
All sequences in [500000, 600000] are calculated up to a new limit: Generally C80. There are now 971 OE-sequences in this interval.

January 2003:
All sequences in [600000, 700000] are calculated up to a new limit: Generally C80. There are now 958 OE-sequences in this interval.

March 2003:
All sequences in [700000, 800000] are calculated up to a new limit: Generally C80. There are now 961 OE-sequences in this interval.
This was the last step - in [1, 10^6] all OE-sequences are calculated up C80 or higher. About 17%  of all numbers in [1, 10^6] - exactly 171251 - are members of these 9486 OE-sequences.

July 2003:
The calculations in [50000, 100000] are finished. All sequences reached a maximum of 10^90 (Creyaufmueller/Stern).


August 2004:
The calculations in (10000, 50000] (Bosma) are still running, too. All sequences  reached a maximum of 10^90, about 50% more than 10^90. 
The calculations ran completely new up to 10^100 (Creyaufmueller). This work is done.

The calculations in (1000, 10000] (Varona/Benito) are still running. All sequences reached a maximum of 10^100. The calculations have been stopped.

September  2005:
The calculation in (50000, 100000] will be extended up to C100. This work is finished.

June 2008:
All sequences in [100000, 200000] are calculated up to a new limit: Generally C100. There are now 961 OE-sequences in this interval.

September 2008:

1) At the moment computations  in the sequences  276564660 and 966 are running (Zimmermann/Howell/Creyaufmueller).

2) The statistics of aliquot sequences in [1, 1000000] will be actualized from time to time. The complete results includes the data base C9C30. Here you'll find all information about terminating and open-end sequences. There you will find the first 9digit-number (C9) and the first 30digit-number (C30) of all sequences.

3) A second data  base C60 shows the first 60digit-number of every sequence in [1, 10^6].

4) A third data base C80 shows the first 80digit-number of every sequence in [1, 10^6]. There are combined results from Zimmermann, Varona, Creyaufmueller, Bosma.

5) The calculations in (1000, 10000] will be extended by Christopher Clavier.

6)
The calculation in (200000, 250000] will be extended up to C100. About 40% of these computations are finished.

7) The calculation in (250000, 300000]
will be extended up to C100 by Wieb Bosma. 

8) The first sequence reached the record index 8033.

June 2009:
All OE-Sequences are online for download

Since January 2010:
A group of free workers will do lots of work on aliquot sequences. To avoid multiple work please check the website of Mersenne-Forum for reserved sequences! (200k, 250k] is up to >C100. (500k, 600k] is reserved, (700k, 800k] too.

October 2011:
The first OE-sequence reached more than index 10000 with actually i12320 (C150)

C9C30 - last update: 12-8-2009 C60 - last update: 17-5-2002 C80 - last update: 31-3-2003
Data base C9C30 - 
partition 1 to 100000
 
Data base C60 - 
partition 1 to 100000
Data base C80 - 
partition
Data base C9C30 - 
complete version (ca: 650 kB)
Data base C60 - 
complete version (ca. 372 kB)
Data base C80 - 
complete version (ca. 446 kB)

definition    aliquot sequences      tables    diagrams    databases    links    literature    factorizing    programs    projects    records    start    ©


recent results


Records

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Please compare this table with Clifford Sterns page and his graphs

 

sequence

status

 graph

date

first computed or maximum reached by

             
Record length:

  204828

  i4015: OE-sequence g204838    Creyaufmueller/
Zimmermann
 ( i > 4000 )   635016   i4090: side-sequence to 25968      Creyaufmueller
    227646   i4203: side-sequence to 25968      Creyaufmueller
    652500   i4208: side-sequence to 25968      Creyaufmueller
      43974   i4126/i4243: OE-sequence    13-06-2004 Bosma / Creyaufmueller
    481900   i4346: terminates in 3   31-08-2009 mataje
      43230   i4356: terminates to 101   g43230  03-12-1999 Bosma
    604440   i4368: OE-sequence   03-01-2003 Creyaufmueller
      98790   i4443: terminates to 13   20-07-2008 Stern/Creyaufmüller
    971088   i4468: OE-sequence   05-10-2000 Creyaufmueller
    216840   i4699: OE-sequence   24-08-2008 Creyaufmueller
    446580   i4736: terminates to 601  g446580 19-04-2002 Creyaufmueller
    321090   i4743: OE-sequence   29-01-2002 Creyaufmueller
      11040   i4746/i4853: OE-sequence   22-12-2001
09-06-2004
Stern
Creyaufmueller
 ( i > 5000 )   643752   i5008: OE-sequence (C83)   06-12-2002 Creyaufmueller
      92898   i5063: terminates to 41   11-03-2010 Stern
    696780   i5293: terminates to 59   09-02-2010 frmky
    921232   i5326: OE-sequence (C103) g921232  03-06-2000  Creyaufmueller
      59232   i5339: OE-sequence (C102)   21-01-2001 Zimmermann/Stern
    115302   i5415: OE-sequence (C100) g115302 24-07-2001 Creyaufmueller
        8760   i5583: OE-sequence (C107)   27-01-2003 Varona
    707016   i5932: terminates in 41   12-11-2010 Batalov
 (i > 6000)   921232   i6358: terminates to 11  g921232-n

12-04-2010

Creyaufmüller/unconnected
    483570   i6491: OES-sequence   23-05-2002 Creyaufmueller
 (i > 7000)   144984   i6531: OE-sequence / i7053   31-08-2007 Creyaufmueller/Schickel
        1578   i7147: OE-sequence (C128 ?) / i7261   22-11-2006 Clavier
    195528   i7955: OE-sequence    2009 Batalov
 (i > 8000)   389508   i7070: OES-sequence (C87) / i7135  (C101) / i8000 (C124) / i8033 /C126) g389508 20-03-2002/
06-12-2006/
21-06-2008
Creyaufmueller/Bosma
Stern/Nelson-Melby
 (i > 9000) ??   314718   OES-sequence: 314718:i6444 = 4788:i6 // 
i9004 (??)
   2009 (?)
14-07-2010
Bosma / Schickel
      11040  i9405: OE-sequence    29-03-2012 unconnected
 (i > 12000)   933436  OE-sequence: 933436: i12320 (C150)    19-10-2011 unconnected
           
Terminating sequences:     64962   i2595 = 7      Zimmermann
        6160   i3026 = 601      g6160 25-12-2001 Varona
      42660   i3057 = 43      Bosma
    849920   i3336 = 7   27-03-1999 Creyaufmueller
      11670   i3534 = 193   05-09-2007 Stern
    483616   i3616 = 31   14-05-2002 Creyaufmueller
      62850   i3973 = 41   04-09-2008 Stern
    481900   i4346 = 3   31-08-2009 mataje
      43230   i4356 = 101   g43230 0 3-12-1999 Bosma
      98790   i4443 = 13   20-08-2008 Stern
    428106   i4717 = 7   20-01-2012 Batalov
    446580   i4736 = 601 g446580 19-04-2002 Creyaufmueller
      92898   i5063 = 41   11-03-2010 Stern
    696780   i5293 = 59   09-02-2010 frmky
    707016   i5932 = 41   12-11-2010 Batalov
    921232   i6358 = 11  g921232-n

12-04-2010

Creyaufmüller/unconnected
    414288   i6584 = 601   24-07-2009 Santos
              
Side-sequences:     42800   i2180 = 4788:i6 = 60564      Bosma
    336048   i2727 = 552: i21 = 772840      Creyaufmueller
    389508   i2919 = 34908:i7 = 113464   20-03-2002 Creyaufmueller
    487140   i2960 = 660: i25 = 14700   10-10-1998 Creyaufmueller
    731520   i3328 = 4116: i7 = 42028   11-02-1999 Creyaufmueller
    644664   i3882 = 37632:i2 = 149716    26-12-2011 bchaffin
    207984   i4215 = 53802: i151 = 1773682   31-12-1011 Batalov
    483570   i4656 = 1920:i81 = 98624   17-05-2002 Creyaufmueller
    438966   i4720 = 5738:i5 = 246584   11-02-2012 unconnected
    314718   314718:i6466 = 4788:i6 = 60564   2009 Schickel/Bosma
          

  

  

Maximum at terminating sequences:       4170: i289  (C84:lg 83.52) terminates in 79   Mai 1997 Bosma
      44922: i1167  C85: lg 84.77) terminates in 41   Nov. 1999 Bosma
      43230: i967  (C91: lg 90.13) terminates in 101    g43230 03-12-1999 Bosma
      16302: i973  (C94: lg 93.85) terminates in 683   31-12-2003 Creyaufmueller
      49218: i973  (C94: lg 93.98) terminates in 59   07-07-2004 Creyaufmueller
   105384: i2014  (C96: lg 95.15) terminates in 1210/1184   02-04-2004 Stern
        6160: i1631  (C96: lg 95.57) terminates in 601      g6160 25-12-2001 Varona/Benito
      62832: i1740  (C100: lg 99.01) terminates in 43   28-06-2002 Stern
      23910: i1216  (C100: lg 99.43) terminates in 1210/1184   22-04-2004 Stern
        3630: i1263  (C100: lg 99.75) terminates in 59      g3630 10-06-2001 Varona/Benito
      56368: i2007  (C102: lg 101.36) terminates in 43   06-04-2005 Stern
    134856: i746  (C102: lg 101.42) terminates in 601  

 15-06-2009

Batalov
     21024: i1059  (C103: lg 102.56) terminates in 1429   30-04-2005 Stern
    754848: i1049  (C104: lg 103.49) terminates in 14891  

31-12-2009

Batalov
    683730: i829  (C105: lg 104.85) terminates in 59  

 08-05-2011

bchaffin
    721980: i886  (C106: lg 105.06) terminates in 59   15-02-2010 Greebley
    767296: i1888  (C106: lg 105.63) terminates in 31   15-05-2011 unconnected
    815730: i989  (C106: lg 105.75) terminates in 1153   26-02-2010 biwema
    477750: i1199  (C106: lg 105.76) terminates in 601   27-06-2011 bchaffin
    557016: i1489  (C107: lg 106.13) terminates in 43   16-03-2011 Schickel
      92898: i3390  (C107: lg 106.29) terminates in 41   11-03-2010 Stern
      90480: i604  (C107: lg 106.41) terminates in 59   15-07-2010 smh
      45984: i841  (C107: lg 106.50) terminates in 11

  g45984

 30-05-2006 Stern
    649248: i1568  (C107: lg 106.79) terminates in 281  

 07-05-2011

Stern

    959916: i602  (C108: lg 107.17) terminates in 43  

17-05-2011

bchaffin
    275892: i1257  (C108: lg 107.22) terminates in 59   26-04-2011 RobertS
      33672: i2069  (C108: lg 107.99) terminates in 41   05-04-2010 unconnected
    930306: i937  (C109: lg 108.09) terminates in 1153   13-05-2011 Stern
      15960: i846  (C109: lg 108.26) terminates in 41   03-05-2007 Stern
    771108: i597  (C109: lg 108.37) terminates in 321329   06-01-2010 unconnected
    877240: i2010  (C109: lg 108.79) terminates in 41   18-06-2011 bchaffin
    757512: i748  (C110: lg 109.04) terminates in 601   20-05-2010 Stern
    673140: i700  (C110: lg 109.12) terminates in 41   18-07-2011 bchaffin
    585600: i598  (C110: lg 109.45) terminates in 37   24-07-2011 unconnected
    752976: i556  (C110: lg 109.49) terminates in 43   16-06-2011 fivemack
    774360: i786  (C110: lg 109.76) terminates in 37   25-09-2011 bchaffin
    829914: i1942  (C111: lg 110.03) terminates in 37   04-04-2010 biwema
    734184: i852  (C111: lg 110.50) terminates in 43   25-09-2011 bchaffin
    368712: i1413  (C112: lg 111.13) terminates in 41   14-07-2011 unconnected
      91008: i1074  (C112: lg 111.53) terminates in 7   05-03-2012 bchaffin
    373152: i375  (C112: lg 111.68) terminates in 601   09-02-2011 RobertS
    543972: i1135  (C112: lg 111.69) terminates in 37   21-11-2011 bchaffin
      98790: i3257  (C112: lg 111.96) terminates in 13    20-07-2008   Stern
    734760: i836  (C113: lg 112.06) terminates in 191   02-09-2012 Winslow
    142764: i1708  (C113: lg 112.14) terminates in 11   13-11-2011 bchaffin
    224560: i1238  (C113: lg 112.15) terminates in 7   09-01-2012 bchaffin
    590556: i2314  (C113: lg 112.25) terminates in 59   01-10-2011 RobertS
    266224: i683  (C113: lg 112.61) terminates in 1093   22-08-2011 bchaffin
    858180: i2818  (C113: lg 112.75) terminates in 59   02-10-2011 bchaffin
    940470: i3600  (C113: lg 112.95) terminates in 59   11-03-2012 unconnected
      11670: i1067  (C113: lg 112.96) terminates in 193   05-09-2007 Stern
    428106: i1880  (C114: lg 113.44) terminates in 7   20-01-2012 Batalov
    712068: i1935  (C114: lg 113.78) terminates in 43   12-02-2012 bchaffin
    555084: i903  (C114: lg 113.84) terminates in 59   17-07-2012 Batalov
    587994: i3177  (C115: lg 114.07) terminates in 5431   18-02-2012 unconnected
    133938: i3015  (C115: lg 114.08) terminates in 601   28-03-2012 unconnected
    507924: i686  (C115: lg 114.17) terminates in 43   15-09-2009 Greebley
    151752: i1175  (C115: lg 114.67) terminates in 59    21-06-2009 Greebley
    834216: i1507  (C115: lg 114.99) terminates in 601   15-04-2010 Batalov
    701184: i783  (C116: lg 115.81) terminates in 73   28-11-2011 bchaffin
    306912: i381  (C116: lg 115.94) terminates in 41   09-07-2011 Batalov
    417600: i532  (C117: lg 116.35) terminates in 43   19-02-2012 Batalov
      58374: i910  (C118: lg 117.37) terminates in 601   29-12-2012 fivemack
      98616: i1041  (C118: lg 117.41 terminates in 43    03-12-2009 Stern
    677430: i659  (C118: lg 117.49 terminates in 59   07-09-2012 Winslow
    163716: i929  (C118: lg 117.91) terminates in 37    12-06-2009

Schickel

    629718: i1331  (C119: lg 118.07) terminates in 41   16-02-2014 unconnected
    327132: i3764  (C119: lg 118.54) terminates in 41   15-02-2013 fivemack
   109128: i1435  (C119: lg 118.75) terminates in 601   14-04-2013 fivemack
    933870: i3165  (C120: lg 119.17) terminates 181   10-12-2012 Batalov
      19494: i1547  (C120: lg 119.92) terminates in 37   29-11-2012 fivemack
    259896: i591  (C121: lg 120.17) terminates in 1210/1184    02-12-2013 Batalov
    770580: i2992  (C121: lg 120.30) terminates in 7   05-02-2013 unconnected
    738288: i1263   (C121: lg 120.72) terminates in 41  

17-03-2013

unconnected

      62850: i3227  (C121: lg 120.98) terminates in 41   04-09-2008 Stern
    856710: i1264  (C123: lg 122.19) terminates in 41   18-12-2013 unconnected
      99240: i763  (C123: lg 122.27) terminates in 397   08-07-2012 Batalov
    331202: i1357  (C123: lg 122.44) terminates in 43  

 20-04-2013

firejuggler
    993438: i1090  (C123: lg 122.62) terminates in 1741  

 20-05-2013

Jatheski
    228522: i2852  (C123: lg 122.65) terminates in 109  

 12-05-2014

Sergiosi
    707016: i3396  (C124: lg 123.81) terminates in 41   12-11-2010 Batalov
    921232: i5510  (C127: lg 126.67) terminates in 11   12-04-2010 Creyaufmüller/unconnected
           
Maximum at OE-sequences:       1134: i2265   C134

  g1134

 30-01-2004  Zimmermann/
Creyaufmueller
          552: i902   C147        g552  15-05-2005 Zimmermann
          276: i1567   C149        g276 11-06-2006 Zimmermann
       3432: i1098   C160    20-05-2006 Clavier
       4788: i2529   C172   21-04-2010 Clavier
       3270: i677   C188   19-02-2013 Clavier
          

  

Maximum at OES-sequences:     72288: i1144  (C90: lg 89.15) - OES to 11408   17-07-2002 Stern
    532530: i1156  (C109: lg 108.88) OES to 10532   07-07-2011 bchaffin
    594228: i969  (C110: lg 109.46) OES to 38208   15-11-2011 bchaffin
    644664: i779  (C112: lg 111.46) OES to 37   26-12-2011 bchaffin
    207984: i4215  (C112: lg 111.54) - OES to 53802   31-12-2011 Batalov
    920468: i1041  (C114: lg 113.57) - OES to 163716   29-03-2012 Batalov
    438966: i4720  (C116: lg 115.63) OES to 5748   11-02-2012 unconnected
    448476: i862  (C117: lg 116.29) - OES to 1074   01-06-2013 Batalov
    273540: i1158  (C117: lg 116.88) - OES to 3876   12-07-2013 unconnected
   346848: i1314  (C119: lg 118.27) - OES to 8844   27-05-2013 Batalov
    574344: i1041  (C124: lg 123.46) OES to 5748   17-09-2012 Stern
Greatest descent:       1578  From C110 down to C5

 g1578

 20-02-2005  Clavier

 

Note to aliquot sequences with record length: 
The three side-sequences to 25968 are linked at different points:
227646:i14 = 652500:i2 = 2372658 and 820584:i2 = 635016:i3 and 635016:i962 = 227646:i1094 = 230456;532
25968:i10 = 196188:i409 = 635016:i978 = 652500:i1100 = 227646:i1112.
The sequence 921232 is the first well known aliquot sequence with more than 5000 iterations.  
The sequence 389508 is the first well known aliquot sequence with more than 8000 iterations; status: 389508:i8033
The calculations of the sequence 115302 have been stopped at index 5415 (C100).

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