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definition aliquot sequences tables diagrams databases links literature factorizing programs projects records start ©
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 x^{m}
 y^{n} = 1 has no other integer solution but 3^{2}
 2^{3} = 1. In May 2002 Preda
Mihailescu gave a proof for this conjecture. On this website you can't find
anything else about these facts.
definition aliquot sequences tables diagrams databases links literature factorizing programs projects records start ©
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 keynumber 17490
ends in an aliquot4cycle. 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
aliquot4cycles is exhaustive. The largest 4cycle 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
startup
number (or keynumber, beginning number) of such a socalled openend sequence (OEsequence 
Offenendkette) is 276 (> labyrinth
of
Chartres). An openend sequence with all side sequences
is called an openend family (OEfamily).
There are 5 openend sequences in the interval [1, 1000] with the key numbers
276, 552,
564, 660 und
966. They are called
Lehmer Five.
There are 81 openendsequences in the interval [1, 10^4].
There are now 898 openend sequences in [1, 10^5] and 9202 OEsequences 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 openend 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 OESequences  > 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 MersenneForum 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 openend
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 (socalled 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 openend sequences were
computed separately with UBASICprograms.
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 OEsequences got the smallest keynumber comparable to
a targetprime.
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 OEsequences about 2  2.5%. Some sequences terminate, the other were identified
as sidesequences.
At Christmas 2001 Varona terminated the sequence 6160 and 1797 numbers in (1,
10^6] changed their family status.
definition aliquot sequences tables diagrams databases links literature factorizing programs projects records start ©
To get a general idea a graphic presentation of an aliquot
sequence is helpful using a semi logarithm axis, i.e. a linear xaxis beginning
with 0 for the index number of the sequence and the yaxis on a scale of decadic
logarithm for the sum of the proper divisors.
So there is a function f: N(n) > log_{10} 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) openend 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)
up 840 976950 980460 17490 2856 276 1578 down
976950  ending in perfect number (6)
up 840 976950 980460 17490 2856 276 1578 down
980460  ending in an amicable pair (2620/2924)
up 840 976950 980460 17490 2856 276 1578 down
17490  ending in an aliquot 4cycle
up 840 976950 980460 17490 2856 276 1578 down
2856  ending in the aliquot 28cycle
up 840 976950 980460 17490 2856 276 1578 down
276  openend sequence
up 840 976950 980460 17490 2856 276 1578 down
1578  OEsequence with deep minimum
(for greater picture click here or above)
up 840 976950 980460 17490 2856 276 1578 down
definition aliquot sequences tables diagrams databases links literature factorizing programs projects records start ©
1) The data of the aliquot sequences in [1, 1000000] includes the data base
C9C30.
Here you'll find all information about terminating and
openend sequences. There you will find the first 9digitnumber (C9) and the
first 30digitnumber (C30) of all sequences. With this data it will be possible
to identify sidesequences.
2) A second data base C60
shows the first 60digitnumber of every sequence. It will be possible to
identify some more sidesequences.
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 OEsequence (the entry is the
negative keynumber)
With this categories you can sort all entries easily.
There are only entries of such sequences which were computed
with UBASICprograms in the
basematrix. It's easy to change entries in such a matrix. Its easy to
calculate a basematrix, too. To change a totally calculated record file is more
difficult because not all sidesequences can be changed.
definition aliquot sequences tables diagrams databases links literature factorizing programs projects records start ©
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 ©
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", 1995^{1}, 176 p. / 1997^{2}, 262
p. / 2000^{3}, 327 p.
ISBN 3980103226.
Here you will find a great bibliography and commented programs
for PCs with complete source code. The following bibliographylink shows the
actual
mathematical part of the 3rd edition.
An interesting website you will find by Scott Contini's FactorWorld. There are downloadpossibilities for many papers.
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 OEsequences 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 timeconsuming.
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 ECMmethod, 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 100digitnumber 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 nfsPapers.
Good information you will find in mathworld,
too.
An interesting site for a gnfsimplementation: GGNFS.
There is one method to run GGNFS in a DOSWindow 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 ZIPfile: 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 C111cofactor 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 C307factorization
(C200factorisation in May 2005), (C174factorisation
in December 2003),
( C158factorisation 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.
definition aliquot sequences tables diagrams databases links literature factorizing programs projects records start ©
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.
Other Programs
Comment:
ALIQUOT is written by Ivo
Duentsch
in Turbo Pascal and available in its original version at the ftpserver. 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 "BorlandTurboPascalruntimeerror
200".
The UBASICprograms 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 datafiles (*.ELF).
In December 2001 Clifford Stern wrote a helpful program to find out the
confluencepoint of sidesequences. A second program converts SQfiles into
*.ELFfiles.
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/Windowscomputer and programs for this machines. Paul
Zimmermann
gives a lot of information about UNIXcomputers 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  WikipediaGNFS  MIRACL  Michael Scott 
ECC Tutorial  MSIEVE  Richard Brent's factor tables 
Elliptic Curves  Wilfried Keller  Faktortafeln  
CryptoWorld  
Factorization Announcements  
numbertheory.org/ntw/N4.html  links  
Primzahlen  prime numbers  
Prime Links ++  
Primzahlseite 
definition aliquot sequences tables diagrams databases links literature factorizing programs projects records start ©
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.
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 OEsequences in this interval.
May 2001:
All sequences in [800000, 900000] are calculated up to a new limit:
Generally C80. There are now 982 OEsequences in this interval.
October 2001:
All sequences in [100000, 200000] are calculated up to a new limit:
Generally C80. There are now 975 OEsequences in this interval.
January 2002:
All sequences in [200000, 300000] are calculated up to a new limit:
Generally C80. There are now 938 OEsequences in this interval.
March 2002:
All sequences in [300000, 400000] are calculated up to a new limit:
Generally C80. There are now 877 OEsequences in this interval.
May 2002:
All sequences in [400000, 500000] are calculated up to a new limit:
Generally C80. There are now 917 OEsequences in this interval.
July 2002:
All sequences in [500000, 600000] are calculated up to a new limit:
Generally C80. There are now 971 OEsequences in this interval.
January 2003:
All sequences in [600000, 700000] are calculated up to a new limit:
Generally C80. There are now 958 OEsequences in this interval.
March 2003:
All sequences in [700000, 800000] are calculated up to a new limit:
Generally C80. There are now 961 OEsequences in this interval.
This was the last step  in [1, 10^6] all OEsequences are calculated up
C80 or higher. About 17% of all numbers in [1, 10^6]  exactly 171251  are
members of these 9486 OEsequences.
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 OEsequences 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
openend sequences. There you will find the first 9digitnumber (C9) and the
first 30digitnumber (C30) of all sequences.
3) A second data base C60
shows the first 60digitnumber of every sequence in [1, 10^6].
4) A third data base C80 shows the first
80digitnumber 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 OESequences 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 MersenneForum
for reserved sequences! (200k, 250k] is up to >C100. (500k, 600k] is reserved,
(700k, 800k] too.
October 2011:
The first OEsequence reached more than index 10000 with actually i12320 (C150)
C9C30  last update: 1282009  C60  last update: 1752002  C80  last update: 3132003 
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 ©
Please compare this table with Clifford Sterns page and his graphs
sequence 
status 
graph 
date 
first computed or maximum reached by 

Record length:  i4015: OEsequence  g204838  Creyaufmueller/ Zimmermann 

( i > 4000 )  635016  i4090: sidesequence to 25968  Creyaufmueller  
227646  i4203: sidesequence to 25968  Creyaufmueller  
652500  i4208: sidesequence to 25968  Creyaufmueller  
43974  i4126/i4243: OEsequence  13062004  Bosma / Creyaufmueller  
481900  i4346: terminates in 3  31082009  mataje  
43230  i4356: terminates to 101  g43230  03121999  Bosma  
604440  i4368: OEsequence  03012003  Creyaufmueller  
98790  i4443: terminates to 13  20072008  Stern/Creyaufmüller  
971088  i4468: OEsequence  05102000  Creyaufmueller  
216840  i4699: OEsequence  24082008  Creyaufmueller  
446580  i4736: terminates to 601  g446580  19042002  Creyaufmueller  
321090  i4743: OEsequence  29012002  Creyaufmueller  
11040  i4746/i4853: OEsequence  22122001 09062004 
Stern Creyaufmueller 

( i > 5000 )  643752  i5008: OEsequence (C83)  06122002  Creyaufmueller  
92898  i5063: terminates to 41  11032010  Stern  
696780  i5293: terminates to 59  09022010  frmky  
921232  i5326: OEsequence (C103)  g921232  03062000  Creyaufmueller  
59232  i5339: OEsequence (C102)  21012001  Zimmermann/Stern  
115302  i5415: OEsequence (C100)  g115302  24072001  Creyaufmueller  
8760  i5583: OEsequence (C107)  27012003  Varona  
707016  i5932: terminates in 41  12112010  Batalov  
(i > 6000)  921232  i6358: terminates to 11  g921232n 
12042010 
Creyaufmüller/unconnected 
483570  i6491: OESsequence  23052002  Creyaufmueller  
(i > 7000)  144984  i6531: OEsequence / i7053  31082007  Creyaufmueller/Schickel  
1578  i7147: OEsequence (C128 ?) / i7261  22112006  Clavier  
195528  i7955: OEsequence  2009  Batalov  
(i > 8000)  389508  i7070: OESsequence (C87) / i7135 (C101) / i8000 (C124) / i8033 /C126)  g389508  20032002/ 06122006/ 21062008 
Creyaufmueller/Bosma Stern/NelsonMelby 
(i > 9000) ??  314718  OESsequence:
314718:i6444 = 4788:i6 // i9004 (??) 
2009 (?) 14072010 
Bosma / Schickel  
11040  i9405: OEsequence  29032012  unconnected  
(i > 12000)  933436  OEsequence: 933436: i12320 (C150)  19102011  unconnected  
Terminating sequences:  64962  i2595 = 7  Zimmermann  
6160  i3026 = 601  g6160  25122001  Varona  
42660  i3057 = 43  Bosma  
849920  i3336 = 7  27031999  Creyaufmueller  
11670  i3534 = 193  05092007  Stern  
483616  i3616 = 31  14052002  Creyaufmueller  
62850  i3973 = 41  04092008  Stern  
481900  i4346 = 3  31082009  mataje  
43230  i4356 = 101  g43230  0 3121999  Bosma  
98790  i4443 = 13  20082008  Stern  
428106  i4717 = 7  20012012  Batalov  
446580  i4736 = 601  g446580  19042002  Creyaufmueller  
92898  i5063 = 41  11032010  Stern  
696780  i5293 = 59  09022010  frmky  
707016  i5932 = 41  12112010  Batalov  
921232  i6358 = 11  g921232n 
12042010 
Creyaufmüller/unconnected  
414288  i6584 = 601  24072009  Santos  
Sidesequences:  42800  i2180 = 4788:i6 = 60564  Bosma  
336048  i2727 = 552: i21 = 772840  Creyaufmueller  
389508  i2919 = 34908:i7 = 113464  20032002  Creyaufmueller  
487140  i2960 = 660: i25 = 14700  10101998  Creyaufmueller  
731520  i3328 = 4116: i7 = 42028  11021999  Creyaufmueller  
644664  i3882 = 37632:i2 = 149716  26122011  bchaffin  
207984  i4215 = 53802: i151 = 1773682  31121011  Batalov  
483570  i4656 = 1920:i81 = 98624  17052002  Creyaufmueller  
438966  i4720 = 5738:i5 = 246584  11022012  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  03121999  Bosma  
16302: i973  (C94: lg 93.85) terminates in 683  31122003  Creyaufmueller  
49218: i973  (C94: lg 93.98) terminates in 59  07072004  Creyaufmueller  
105384: i2014  (C96: lg 95.15) terminates in 1210/1184  02042004  Stern  
6160: i1631  (C96: lg 95.57) terminates in 601  g6160  25122001  Varona/Benito  
62832: i1740  (C100: lg 99.01) terminates in 43  28062002  Stern  
23910: i1216  (C100: lg 99.43) terminates in 1210/1184  22042004  Stern  
3630: i1263  (C100: lg 99.75) terminates in 59  g3630  10062001  Varona/Benito  
56368: i2007  (C102: lg 101.36) terminates in 43  06042005  Stern  
134856: i746  (C102: lg 101.42) terminates in 601 
15062009 
Batalov  
21024: i1059  (C103: lg 102.56) terminates in 1429  30042005  Stern  
754848: i1049  (C104: lg 103.49) terminates in 14891 
31122009 
Batalov  
683730: i829  (C105: lg 104.85) terminates in 59 
08052011 
bchaffin  
721980: i886  (C106: lg 105.06) terminates in 59  15022010  Greebley  
767296: i1888  (C106: lg 105.63) terminates in 31  15052011  unconnected  
815730: i989  (C106: lg 105.75) terminates in 1153  26022010  biwema  
477750: i1199  (C106: lg 105.76) terminates in 601  27062011  bchaffin  
557016: i1489  (C107: lg 106.13) terminates in 43  16032011  Schickel  
92898: i3390  (C107: lg 106.29) terminates in 41  11032010  Stern  
90480: i604  (C107: lg 106.41) terminates in 59  15072010  smh  
45984: i841  (C107: lg 106.50) terminates in 11  30052006  Stern  
649248: i1568  (C107: lg 106.79) terminates in 281 
07052011 
Stern 

959916: i602  (C108: lg 107.17) terminates in 43 
17052011 
bchaffin  
275892: i1257  (C108: lg 107.22) terminates in 59  26042011  RobertS  
33672: i2069  (C108: lg 107.99) terminates in 41  05042010  unconnected  
930306: i937  (C109: lg 108.09) terminates in 1153  13052011  Stern  
15960: i846  (C109: lg 108.26) terminates in 41  03052007  Stern  
771108: i597  (C109: lg 108.37) terminates in 321329  06012010  unconnected  
877240: i2010  (C109: lg 108.79) terminates in 41  18062011  bchaffin  
757512: i748  (C110: lg 109.04) terminates in 601  20052010  Stern  
673140: i700  (C110: lg 109.12) terminates in 41  18072011  bchaffin  
585600: i598  (C110: lg 109.45) terminates in 37  24072011  unconnected  
752976: i556  (C110: lg 109.49) terminates in 43  16062011  fivemack  
774360: i786  (C110: lg 109.76) terminates in 37  25092011  bchaffin  
829914: i1942  (C111: lg 110.03) terminates in 37  04042010  biwema  
734184: i852  (C111: lg 110.50) terminates in 43  25092011  bchaffin  
368712: i1413  (C112: lg 111.13) terminates in 41  14072011  unconnected  
91008: i1074  (C112: lg 111.53) terminates in 7  05032012  bchaffin  
373152: i375  (C112: lg 111.68) terminates in 601  09022011  RobertS  
543972: i1135  (C112: lg 111.69) terminates in 37  21112011  bchaffin  
98790: i3257  (C112: lg 111.96) terminates in 13  20072008  Stern  
734760: i836  (C113: lg 112.06) terminates in 191  02092012  Winslow  
142764: i1708  (C113: lg 112.14) terminates in 11  13112011  bchaffin  
224560: i1238  (C113: lg 112.15) terminates in 7  09012012  bchaffin  
590556: i2314  (C113: lg 112.25) terminates in 59  01102011  RobertS  
266224: i683  (C113: lg 112.61) terminates in 1093  22082011  bchaffin  
858180: i2818  (C113: lg 112.75) terminates in 59  02102011  bchaffin  
940470: i3600  (C113: lg 112.95) terminates in 59  11032012  unconnected  
11670: i1067  (C113: lg 112.96) terminates in 193  05092007  Stern  
428106: i1880  (C114: lg 113.44) terminates in 7  20012012  Batalov  
712068: i1935  (C114: lg 113.78) terminates in 43  12022012  bchaffin  
555084: i903  (C114: lg 113.84) terminates in 59  17072012  Batalov  
587994: i3177  (C115: lg 114.07) terminates in 5431  18022012  unconnected  
133938: i3015  (C115: lg 114.08) terminates in 601  28032012  unconnected  
507924: i686  (C115: lg 114.17) terminates in 43  15092009  Greebley  
151752: i1175  (C115: lg 114.67) terminates in 59  21062009  Greebley  
834216: i1507  (C115: lg 114.99) terminates in 601  15042010  Batalov  
701184: i783  (C116: lg 115.81) terminates in 73  28112011  bchaffin  
306912: i381  (C116: lg 115.94) terminates in 41  09072011  Batalov  
417600: i532  (C117: lg 116.35) terminates in 43  19022012  Batalov  
58374: i910  (C118: lg 117.37) terminates in 601  29122012  fivemack  
98616: i1041  (C118: lg 117.41 terminates in 43  03122009  Stern  
677430: i659  (C118: lg 117.49 terminates in 59  07092012  Winslow  
163716: i929  (C118: lg 117.91) terminates in 37  12062009 
Schickel 

629718: i1331  (C119: lg 118.07) terminates in 41  16022014  unconnected  
327132: i3764  (C119: lg 118.54) terminates in 41  15022013  fivemack  
109128: i1435  (C119: lg 118.75) terminates in 601  14042013  fivemack  
933870: i3165  (C120: lg 119.17) terminates 181  10122012  Batalov  
19494: i1547  (C120: lg 119.92) terminates in 37  29112012  fivemack  
259896: i591  (C121: lg 120.17) terminates in 1210/1184  02122013  Batalov  
770580: i2992  (C121: lg 120.30) terminates in 7  05022013  unconnected  
738288: i1263  (C121: lg 120.72) terminates in 41 
17032013 
unconnected 

62850: i3227  (C121: lg 120.98) terminates in 41  04092008  Stern  
856710: i1264  (C123: lg 122.19) terminates in 41  18122013  unconnected  
99240: i763  (C123: lg 122.27) terminates in 397  08072012  Batalov  
331202: i1357  (C123: lg 122.44) terminates in 43 
20042013 
firejuggler  
993438: i1090  (C123: lg 122.62) terminates in 1741 
20052013 
Jatheski  
228522: i2852  (C123: lg 122.65) terminates in 109 
12052014 
Sergiosi  
707016: i3396  (C124: lg 123.81) terminates in 41  12112010  Batalov  
921232: i5510  (C127: lg 126.67) terminates in 11  12042010  Creyaufmüller/unconnected  
Maximum at OEsequences:  1134: i2265  C134  30012004  Zimmermann/ Creyaufmueller 

552: i902  C147  g552  15052005  Zimmermann  
276: i1567  C149  g276  11062006  Zimmermann  
3432: i1098  C160  20052006  Clavier  
4788: i2529  C172  21042010  Clavier  
3270: i677  C188  19022013  Clavier  


Maximum at OESsequences:  72288: i1144  (C90: lg 89.15)  OES to 11408  17072002  Stern  
532530: i1156  (C109: lg 108.88) OES to 10532  07072011  bchaffin  
594228: i969  (C110: lg 109.46) OES to 38208  15112011  bchaffin  
644664: i779  (C112: lg 111.46) OES to 37  26122011  bchaffin  
207984: i4215  (C112: lg 111.54)  OES to 53802  31122011  Batalov  
920468: i1041  (C114: lg 113.57)  OES to 163716  29032012  Batalov  
438966: i4720  (C116: lg 115.63) OES to 5748  11022012  unconnected  
448476: i862  (C117: lg 116.29)  OES to 1074  01062013  Batalov  
273540: i1158  (C117: lg 116.88)  OES to 3876  12072013  unconnected  
346848: i1314  (C119: lg 118.27)  OES to 8844  27052013  Batalov  
574344: i1041  (C124: lg 123.46) OES to 5748  17092012  Stern  
Greatest descent:  1578  From C110 down to C5  20022005  Clavier 
Note to aliquot sequences with record length:
The three
sidesequences 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).
definition aliquot sequences tables diagrams databases links literature factorizing programs projects records start ©
Email: Wolfgang.Creyaufmueller@tonline.de
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