ALIQUOT PAGE - Prime families

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

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 x^m - 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.

Aliquot Sequences

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 (February 2015) we know 237 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 226 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 Five. There are 81 open-end-sequences in the interval [1, 10^4].
There are now 898 open-end sequences in [1, 10^5] and 9190 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].

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.

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.

Diagrams

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.

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

Diagrams of Genealogy

Databases

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

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

Internet - links

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.

Literature

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

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.

Programs

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. All programmes will run in Windows 7 in a DOS-Box with less comfort.

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)

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.

School projects

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

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 276, 564, 660 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.

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)

recent results

Records

	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) OE-sequence: 933436: i12516 (C183)		19-10-2011 29-05-2016	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
	945360: i839	(C120: lg 119.08) terminates in37		22-11-2014	unconnected
	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
	927468: i1304	(C120: lg 119.96) terminates in 41		05-06-2014	f1pokerspeed
	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
	843528: i1533	(C121: lg 120.59) terminates in 1511		17-02-2015	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
	675204: i1760	(C122: lg 121.85) terminates in 2620/2924		30-12-2014	RobertS
	860076: i1027	(C122: lg 121.90) terminates in 7		23-04-2015	unconnected
	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
	966444: i919	(C123: lg 122.67) terminates in 1153		26-02-2015	unconnected
	707016: i3396	(C124: lg 123.81) terminates in 41		12-11-2010	Batalov
	643448: i900	(C125: lg 124.01) terminates in 601		30-12-2014	Bataloc
	706200: i1012	(C125: lg 124.08) terminates in 59		31-03-2015	unconnected
	660212: i1188	(C125: lg 124.71) terminates in 73		25-09-2014	Batalov
	921232: i5510	(C127: lg 126.67) terminates in 11		12-04-2010	Creyaufmüller/unconnected
	407856: i955	(C128: lg 127.71) terminates in 41		30-11-2014	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 4788: i5232	C172 C195		21-04-2010 20-05-2014	Clavier Propper
	3270: i677	C188		19-02-2013	Clavier
	2340: i750	C200		28-01-2016	Womack

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
	337662: i917	(C122: lg 121.86) - OES to 30352		20-04-2015	fivemack
	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).

interval	number of OE- sequences	limits of computation	computed by

[1, 1000]	5	> 10^186	C/Z
[1, 10000]	81	> 10^151	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]	850	> 10^80 / 10^100	C/B
(400000, 500000]	882	> 10^80	C
(500000, 600000]	945	> 10^80	C
(600000, 700000]	932	> 10^80	C
(700000, 800000]	919	> 10^80 / 10^100	C
(800000, 900000]	952	> 10^80	C
(900000, 10^6]	951	> 10^80	C

[1, 10^6]	9190	> 10^80 / 10^100
Download all OE-Sequences		> 10^80 / 10^100

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

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

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)

Aliquot Sequences

Definition - Catalan Conjecture

Aliquot Sequences

Diagrams

Diagrams of Genealogy

Databases

ALIQUOT and corresponding record files

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Literature

Verlagsbuchhandlung Creyaufmüller

Factorizing great Integers

Programs

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)

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last update: 29-05-2016 © Copyright by author. Imprint