Note: Aside from this note, this is an almost exact duplicate of what used to be this web page, written by Clifford Stern and archived by the Wayback Machine on July 17th, 2011. If the link works for you, please tell me so that I may take down this page. Some of the links here are to other pages on Clifford's site, which I don't have, so those links won't work.

Aliquot Sequences from the Trenches

First, there were perfect numbers. Then came amicable pairs. Those were followed by sociable numbers. This progression led some to question the fate of sequences in which each member was the sum of the aliquot parts of its predecessor, where the aliquot parts of n consist of the divisors of n excluding n itself. This is expressed mathematically by use of the sigma function: (n) equals the sum of all the divisors of the integer n. For each ni of an aliquot sequence, its successor ni+1 = (ni) - ni. The Catalan-Dickson Conjecture theorized that each of these sequences would either collapse to 1 or enter a cycle consisting of one or more numbers. However, the advent of computers has revealed empirically that this conjecture is almost certainly false. In particular, an aliquot sequence beginning with a randomly chosen even number would, in all probability, be expected to increase without bound. This is the assessment of Richard Guy and John Selfridge writing in 1974 and published the next year in the journal Mathematics of Computation. Nevertheless, these sequences continue to be of interest, and the conjecture is considered the best known iteration problem in number theory. Here are the principle drivers that serve to propel the sequences upward:

2 * 3
22 * 7
23 * 3 * 5
24 * 31
25 * 3 * 7
26 * 127

Escape from any of these occurs only under special circumstances (see analysis). One might intuitively think that once a sequence does lose its driver, it would then fall of its own weight, plunging all the way to 1 or ending up in a cycle. But this is rarely the case. After escape from a driver, the next step needed for possible termination would be aquisition of the downdriver (2 *). However, it is far more likely that an upward-propelling guide, driver or the digit '3' would be obtained. As a consequence, these sequences are subject to a natural upward trend.

MAJOR PROJECTS

The biggest project by far was the ambitious effort by Wolfgang Creyaufmüller to compute all sequences that begin with numbers less than 1,000,000 to at least 80 digits. This multi-year endeavor produced a steady stream of interesting results such as terminating sequences, confluences, and exceptionally long sequences. Its completion was announced on April 1, 2003, and the results are now available to the research community on his recently constructed download page.

Paul Zimmermann, the principle investigator of the Lehmer Five, computed each sequence in [1, 10^5] and extended those in [50,000, 10^5] to 80 digits. He made significant contributions to the GNU multiple precision arithmetic library, and his ECMNET Project has produced a series of ever more efficient versions of the elliptical curve factoring method.

Wieb Bosma did pioneering work in [10^4, 50,000], computing each member to at least 90 digits. His results for terminating sequences held up for a remarkably long period of time until Wolfgang Creyaufmüller broke the impasse by terminating 16302 on December 31, 2003. He is now working on extending sequences in [250000, 400000] to 100 digits.

Juan L. Varona and Manuel Benito worked intensively in [10^3, 10^4], computing each to 100 digits and beyond. On June 10, 2001, they terminated 3630, a result which reigned supreme for nearly four years in terms of the maximum size reached for a terminating sequence: 100 digits at index 1263 (5.64 x 10^99). At this site can also be found 3630finishes.pdf, a comprehensive introduction to the subject by multiple authors. In addition, they provide Advances in Aliquot Sequences which was published in the journal Mathematics of Computation.

Christophe Clavier, along with a high-powered team of associates took over work in the interval 1464 to 9852. The result was a remarkable series of driver escapes and downdriver aquisitions. However, they have been very unlucky in failing to obtain any new terminating sequences from their 32 downdriver runs that began in triple digits.

Wolfgang Creyaufmüller and I continued the efforts of Paul Zimmermann in [50,000, 100,000]. My computations were confined to those that were not being driven by one of the main six drivers at the 80-digit mark. The result was one terminating sequence (62832) and one new confluence (72288). Wolfgang followed by working on the remaining ones, and was successful in finding three new terminating sequences, in spite of each having been under the control of a driver at the 80-digit mark.

Wolfgang Creyaufmüller subsequently went to work extending Wieb Bosma's sequences in [10,000, 50,000] to at least 100 digits. Though termination is rare after the 90-digit mark has been reached, he succeeded in finding two new terminating sequences: the aforementioned 16302 and then 49218 near the end of the project.

Wolfgang Creyaufmüller completed a project to extend all sequences in [50,000, 100,000] to at least 100 digits. It produced many interesting results, such as driver escapes and long downdriver runs. However, for the first time, no additional terminating sequences were found.

Wolfgang Creyaufmüller extended the sequences in [100,000, 200,000] to 100 digits. This produced many interesting results, including eleven terminations, one side-sequence and several new confluences.

Members of the Mersenne Forum have extended nearly every sequence in [500,000, 1,000,000] from where Wolfgang Creyaufmüller had left off to at least 100 digits. This produced a long list of new terminations and confluences.

Recent News

Note: results from the frenetic activity in the Mersenne Forum have been provided to me by Frank Schickel

Serge Batalov computed 707016, a sequence with a remarkable number of twists and turns, to the second highest termination at 124 digits. It includes the third longest downdriver run, and is the fourth longest in the termination category with a final index of 5932.

Members of the Mersenne Forum factored the 170-digit composite at index 2549 of 4788 where the sequence had been stuck for several months. It is the record for biggest factorization job for an aliquot sequence using the Number Field Sieve. The sieving was done by Andreas Schindel, Frank Schickel, Ben Chaffin, Sander Hoogendoorn, Ben Buhrow, and Carlos Pinho.

Dmitry Domanov obtained the downdriver at a second highest 147 digits in 11040. His computations have also advanced the sequence to tenth place in the list of longest sequences, where its current index is 6241.

Don Leclair set multiple records by capturing the downdriver at 158 digits while simultaneously escaping from the 2^2 * 7 driver in 3906. It is the highest downdriver acquisition, highest driver escape and highest escape from a driver directly to the downdriver.

Dmitry Domanov became the new king of high terminating sequences by finishing off 921232 on April 12th after it had reached a maximum of 127 digits. Wolfgang Creyaufmüller had originally computed this to 103 digits at index 5326 where it was driven by 2^3 * 3^2 * 5. After advancing it to the 127-digit mark, Dmitry captured the downdriver at 126 digits, which lowered the height to 113. That was followed by the fifth longest downdriver run, producing a reduction in size from 113 to 33. The final index of 6358 makes it the second longest terminating sequence, behind 414288 computed by Ignacio Santos, which ended at index 6584.

98616 terminated after reaching a height of 118 digits on November 18th, tying it for what was then second place with 163716 in the overall list. See below for details.

Christophe Clavier set what was then a new record for the highest escape from 2^2 * 7 directly to the downdrver at 124 digits in 4380. This surpassed the previous high of 116 set by Mersenne Forum members in 10212.

Paul Zimmermann set what was then a new record for the highest acqusition of the downdriver at 145 digits in 660 in July of 2006.

Paul Zimmermann set a new record of 146 for the highest escape from the 2^5 * 3 * 7 driver in 552 during the Summer of 2005, greatly exceeding the previously known record of 107 in 271932.

Frank Schickel performed the hat trick of escapes from the rare 2^9 * 3 * 11 * 31 driver by negotiating escapes in all three sequences that had been under control of that driver. First, he set a new record with an escape in 604560 at 126 digits, greatly exceeding the previous high of 55 set by Wolfgang Creyaufmüller in 867690. That was followed by an escape in 479632 at 111 digits, and on October 16, 2009 he broke his own record with an escape in 363270 at 133 digits.

Paul Zimmermann set a new record of 150 for the highest escape from the 2^5 * 3^n * 7 (n>1) guide in 1134 at index 2308, surpassing the previous high of 135 set by Ignacio Santos in 199152 at index 4572.

Jonathan DeMarrais terminated 507924 after it had reached a maximum of 115 digits, equaling the size of his previous result in 151752 discussed below. Beginning where Wolfgang Creyaufmüller left off at 83 digits with the sequence driven by 2 * 3^2, escape came at 86 digits but the 2^3 * 3 driver then increased the size to 104. Lasting stability was finally achieved at the height of 114, followed by an exceptionally long downdriver run from 115 to 54 digits. The finish came with the prime 43 at index 3147.

Serge Batalov broke is own record for the longest single sequence by extending 552150 to 8190 lines. Picking up where Wolfgang Creyaufmüller left off at index 3964, he advanced the sequence to a length once the exclusive province of paired sequences.

Frank Schickel has reproduced the seminal article mentioned above in .pdf format. What Drives an Aliquot Sequence? by Richard Guy and J. L. Selfridge develops the driver concept and argues in favor of an infinitude of unbounded sequences. Consult the Mersenne Forum for information on obtaining a copy (or email me).

Serge Batalov achieved his third terminating sequence in 216810. Wolfgang Creyaufmüller had left off just when it had escaped from a long run of the 2 * 3 driver, where it emerged guided by 2^2 * 3 at 100 digits. Continuing from there, Serge captured the downdriver at 102 digits, and there was little resistance the subsequent plunge to completion.

Jonathan DeMarrais computed 151752 to the third highest terminating sequence by reaching a maximum of 115 digits. Beginning where Wolfgang Creyaufmüller left off at 100 digits with the sequence stable at index 1117, he quickly ran into trouble when the 2^3 * 3 * 5 driver was acquired. The situation improved when the '3' became raised above the exponent of 1, but that also served to accelerate the rise of the sequence up to 115 digits before escape finally came. The downdriver was obtained at the level, and then began a rare series of events hypothesized below in which a rapid collapse ensued with only small interruptions along the way. It ended at index 2025 with the prime 59.

Serge Batalov got his second terminating sequence in 134856 after it had reached a maximum height of 102 digits. His computations began where Wolfgang Creyaufmüller had left off at 100 digits at index 710.

After coming tantalizingly close with 48462 (see below), Frank Schickel finally achieved his first terminating sequence in 163716, and in so doing, made second place in the list of highest terminating sequences at 118 digits. His work began where Wolfgang Creyaufmüller had left off at 100 digits at index 655.

Serge Batalov set a new record for the longest single sequence by extending 195528 beyond the length of 1578, the previous record held by Christophe Clavier for over three years. Beginning where Wolfgang Creyaufmüller left off at index 3974 where 195528 was driven by 2^2 * 7 at 100 digits, Serge computed the sequence to an escape from that driver at 110 digits. The downdriver was then acquired at that level, initiating a drop to 34 digits. The sequence was subsequently driven back to the height of 101 by 2^4 * 31, and after reaching 115 digits, another long decline reduced the size twice to only 12 digits. Eventually, a return of the 2^2 * 7 driver occurred at 28 digits, sending the sequence to the height of 116, where escape was unfortunately to the 2 * 3 driver. It has now reached the index 7972 at 140 digits. 1578 is at index 7261, and also collapsed to the 2 * 3 driver.

Ignacio Santos set a new record for the highest escape from the 2^3 * 3 * 5 driver at 135 digits in 167148, exceeding the previous high of 129 in 162126.

Ignacio Santos acquired the downdriver at 141 digits in 199152, exceeding the previous record of 140 he had previously set in the very same sequence. The earlier downdriver run lasted only 14 lines, with subsequent computations increasing the size from 137 to 141.

Mikael Klasson set a new standard in 139314 for the longest downdriver run by surpassing the records recently set by Markus Tervooren described below. Size was reduced from 131 to 23 for a reduction of 108 digits, and the total length was 565 lines. After getting as low as 17 digits, the 2^2 * 7 driver was acquired at size 18, and when escape was finally achieved, the sequence had returned to triple digits.

Serge Batalov terminated 191430, thus becoming the first to terminate a six-digit sequence after it had reached triple digits. Beginning where Wolfgang Creyaufmüller had left off at index 1243 where the sequence was at what turned out to be its maximum height of 100, the downdriver was acquired at 99 digits at index 1267, leading to its ultimate termination. This broke a three-way tie at the previous maximum height of 99 for six-digit terminating sequences held by Wieb Bosma in 261306 and Wolfgang Creyaufmüller in 108072 and 133596.

Ignacio Santos obtained the downdriver in 199152 at a new record height of 140, surpassing the previous high of 134 set by Markus Tervooren in 162126. In addition, it was obtained while simultaneously losing the '3', breaking the previous record of 133 digits recently set by Mikael Klasson in 103920.

Markus Tervooren completed a remarkable downdriver run in 162126, breaking previous records by a substantial margin. The reduction is size was 101 and the length was 536, surpassing the previous records of 88 an 436 respectively, set by Christophe Clavier in 6822.

Max Dettweiler posting at mersenneforum.org set a new record for the highest escape from the 2^6 * 127 driver at 105 digits in 102264, edging out the previous record of 104 set by Wolfgang Creyaufmüller in 82728. Of special interest was that the escape was directly to the downdriver, exceeding the previous high of 76 digits for such a transition from this driver held by Wolfgang in 838476.

Dennis Langdeau set a new record for the highest escape from the 2^4 * 31 driver at 129 digits in 5778. The previous high was 119 in 6822.

Christophe Clavier broke the record for the highest escape from the 2^5 * 7 guide at 143 digits in 3366. The previous record was 119 digits in 1512.

Wieb Bosma posted a remarkable length result for 314718, and Frank Schickel reconstructed the sequence after spotting it on Wieb's web page. Subsequent analysis revealed that it had merged with 4788 (314718:i6466 = 4788:i6 = 60564) so that the combined 314718/4788 set a new record as the longest sequence pair, now at index 9063. That dropped 389508/34908 to second place at index 8034 and 483570/1920 to third among paired sequences at index 6865.

Frank Schickel has written a valuable new graphing program for aliquot sequences. Programmed in Delphi using Steema software, sequences in virtually any format can be loaded and then saved as a GIF graphic file. I've added a graphics page with especially interesting examples, most of which are discussed or mentioned on this page. Others include 76686 and the terminating 133596, both by Wolfgang Creyaufmüller.

Members of mersenneforum.org in the Aliquot-sequence section set what was then a new record of 116 digits for the highest escape from 2^2 * 7 directly to the downdriver in 10212 at index 1614. The previous record was 111 digits in 1734 at index 1484. 10metreh began by extending the work of Wieb Bosma and Wolfgang Creyaufmüller, and Andreas Schindel began the downdriver run, eventually ending at 82 digits. The 2 * 3 driver was acquired at the height of 97, sending the sequence to its current size of 138.

Older News

62850 set what was then a new record for the highest terminating sequence when it finished on September 3rd, 2008, exactly one week after aquiring the downdriver at 121 digits. See below for details.

Late in the morning of July 20th, 2008, 98790 ended in the prime 13. It was second in length (4443) and now third in maximum height (112) for terminating sequences. 446580 continues in first place in the category of longest terminating sequence at 4736.

Christophe Clavier broke multiple records with a downdriver run of historic dimensions in 6822: it was the longest at 436 lines and produced the biggest reduction in size of 88, falling from 129 to 41 digits. The previous records were 386 and 81 respectively, in 33960. In addition, it reached a new record height (137) for a sequence that would later aquire the downdriver. Christophe had earlier broken the record for the highest escape from the 2^3 * 5 guide at 133 digits in this sequence, surpassing the previous record of 119 set by Igor Schein in 7890.

Christophe Clavier escaped the 2 * 3 driver at a new record height of 139 in 1464, surpassing the previous record of 125 which he also set for 3876. The sequence is now guided by 2^4 * 3^2.

Frank Schickel obtained the downdriver in 144984 at 128 digits, then the fifth highest overall. Size was reduced to 95, where the downdriver was lost directly to the 2^3 * 5 guide. It achieved the milestone of 7000 lines and ended up driven by 2^5 * 3 * 7 at index 7010 and height 132.

Frank Schickel obtained the downdriver in 47728 at 122 digits, a new record height for sequences that began in the five-digit range. The previous high was 115 for 48462 and 82728. The downdriver was lost at the height of 112, and the sequence is now driven by 2^3 * 3^2 at 133 digits.

Christophe Clavier escaped the 2^3 * 3 driver at a new record height of 143 in 2484, surpassing the previous record of 142 which he also set for 1560.

11670 was the 22nd sequence to aquire the downdriver at a size of at least 110 digits, and became the first of these to terminate when it ended in the prime 193 on September 4, 2007. See below for details.

Wolfgang Creyaufmüller extended 144984 from its previous target height of 80 digits at index 1638 to a confluence with 391040 at index 5255: 144984:i5255 = 391040:i6 = 1031680. 391040 was thus reduced to side-sequence status. Wolfgang continued computations of 144984 to 101 digits, where it reached the index 6531, and thus replaced 8760 as the second longest single sequence. 144984 was later computed by Frank Schickel (see above).

Frank Schickel extended 48462 from where I left off to an escape from the 2^4 * 31 driver that had controlled the sequence since its aquisition at height 43. Escape came at 113 digits, the second highest for that driver. He then obtained the downdriver at the size of 115 digits, then a new record height for such aquisition in a sequence beginning in [10^4, 10^6]. The downdriver was lost at 103 digits but was reacquired at height 106 and then again at 108 digits, resulting in a reduction of size to 60. Numerous downdriver runs followed, eventually dropping the sequence twice to only 9 digits. Unfortunately, the recalcitrant 2^6 * 127 driver was acquired at 17 digits, sending the sequence to the height of 123.

Benoît Chevallier-Mames set what was then a new record for the highest aquisition of the downdriver: 133 digits in 9684. This surpassed the previous record of 132 digits set by Christophe Clavier at index 4359 in 1578. The downdriver was lost at size 113, but two more downdriver runs reduced the size all the way to 39. The 2^2 * 7 driver took over directly from there, and has driven the sequence to 135 digits.

Eric Nelson-Melby extended 34908 to index 5088 and height 124. In March of 2002, Wolfgang Creyaufmüller had found 389508 to be a side-sequence of 34908: 389508:i2919 = 34908:i7 = 113464. As a result, the combined sequence 389508/34908 was at a then record index of 8032.

Stable Sequences in [10^4, 10^5]

The following sequences are currently "stable" in that they are not being propelled upward by a driver, guide, or by the presence of that pesky '3':

84822 c100 sz 115  Z/C/W  downdriver run on hold
80244 c115 sz 118  Z/C/W
61452 c100 sz 119  Z/C/J
42798 c120 sz 124  B/C/W
29360 c130 sz 131  B/C/S
23142 c131 sz 133  B/C/S
63660 c134 sz 136  Z/C/S
11040 c115 sz 141  B/C/S/D

B=Bosma, C=Creyaufmüller, D=Dmitry Domanov, J=Jonathan DeMarrais, W=Tom Womack, Z=Zimmermann, S=Stern

The current status of the 822 unresolved sequences that began with five digit numbers is partially given in this text file. Please check the master database for more recent results for certain sequences.

In July, I uploaded every sequence I've worked on since November of 2003 to the master database that had more lines than were already there. I'm currently working on the following list of 13 in the 5-digit range:

21756 23142 29360 35196 38052 41412 52374 55752 56928 63660 75798 81600 85716.

Three are stable (given in the list above), and the others are one possible step from stabiliy. They are of the form 2^2 * 3, 2^4 * 3, 2^3 * 5 or 2^5 * 7 and are in the range 130-136 digits.

Sequences that reached triple digits before terminating

 height sequence date 127 921232 April 12, 2010 Dmitry Domanov 124 707016 November 11, 2010 Serge Batalov 121 62850 September 3, 2008 graph 118 98616 November 18, 2009 118 163716 June 11, 2009 Frank Schickel 115 151752 June 21, 2009 Jonathan DeMarrais 115 507924 September 15, 2009 Jonathan DeMarrais 115 834216 April 15, 2010 Serge Batalov 113 11670 September 4, 2007 112 98790 July 20, 2008 111 829914 April 4, 2010 biwema 110 757512 May 20, 2010 109 15960 May 5, 2007 109 771108 January, 2010 Dmitry Domanov 108 33672 April 5, 2010 Dmitry Domanov 107 45984 May 29, 2006 107 90480 April 15, 2010 Sander Hoogendoorn 107 92898 March 6, 2010 107 534200 September, 2009 Mersenne Forum 106 721980 February, 2010 Jonathan DeMarrais 106 815730 February 26, 2010 biwema 104 754848 December, 2009 Serge Batalov 104 938490 October 31, 2010 103 21024 April 30, 2005 103 334440 November 20, 2010 RobertS 103 812376 February, 2010 Tom Womack 103 894312 August 31, 2010 102 56368 April 5, 2005 102 134856 May 5, 2009 Serge Batalov 102 216810 July 21, 2009 Serge Batalov 101 17130 August 26, 2005 101 197208 March 14, 2010 Serge Batalov 101 791112 January, 2010 Jonathan DeMarrais 101 904386 March 23, 2010 Greg Childers 101 964614 November, 2009 Serge Batalov 100 3630 June 10, 2001 Juan L. Varona and Manuel Benito 100 23910 April 21, 2004 ended in 1210, 1184 100 62832 June 28, 2002 100 191430 May 5, 2009 Serge Batalov 100 461430 December 11, 2010 Ben Chaffin

Highest driver escapes

 driver height sequence index escaped by 2 * 3 139 1464 2248 Christophe Clavier 2^2 * 7 158 3906 1571 Don Leclair 2^3 * 3 144 5778 1047 Christophe Clavier 2^3 * 3 * 5 135 167148 3916 Ignacio Santos 2^4 * 31 129 5778 973 Dennis Langdeau 2^5 * 3 * 7 146 552 900 Paul Zimmermann 2^6 * 127 105 14994   53970 102264 649   567 3140 Jonathan DeMarrais Jonathan DeMarrais Max Dettweiler 2^9 * 3 * 11 * 31 133 363270 1619 Frank Schickel

My results

I've found seventeen terminating sequences and two new confluences:

938490 terminated on October 31st after having reached a maximum size of 104. Wolfgang Creyaufmüller computed this sequence to 81 digits at index 1665 where it was driven by 2^3 * 3. Greg Childers extended it on April 1st to 101 digits where it was guided by 2^2 * 3 at index 1742. I restarted work on October 23rd, and after topping out at 104, it finished on Halloween eight days later.

894312 terminated on August 31st after having reached a maximum size of 103. Wolfgang Creyaufmüller originally computed this sequence to 81 digits where it was driven by 2^2 * 3 * 7 at index 552. Tom Womack continued from there and obtained an escape from that driver directly to the downdriver at the height of 86. He eventually left off at the target size of 100 where it was guided by 2^2 * 3 * 5^2. I built upon those computations beginning on August 28th, and termination came three days later upon reaching a confluence with 6160: 894312:i3098 = 6160:i1 = 11696. That merger added 3000+ lines, resulting in a final index of 6123, making it the third longest terminating sequence. First and second places are 414288 and 921232 discussed above. 6160 along with 3630 are the historic triumphs of Juan L. Varona and Manuel Benito. Those sequences terminated after reaching heights of 96 and 100 respectively in 2001. Remarkably, they continue to be the highest terminating sequences in [1, 10^4].

757512 terminated on May 20th after reaching a maximum height of 110 digits: 757512i2207=601. Originally computed by Wolfgang Creyaufmüller, he acquired the 2^5 * 3^2 * 7 guide at 69 digits, which was still in effect when he left off at index 571 at 82 digits. A Forum member obtained a one line escape four lines later, whereupon it snapped back to something even worse: the 2^5 * 3 * 7 driver. However, an escape from that came at 93 digits, and was left off guided by 2^5 * 3^2 * 5 at 100 digits. I began work on May 15, and stability was achieved just 10 lines later at 103 digits. It continued to run up all the way to 110 digits where the downdriver was obtained. That reduced the size to just 105, but another downdriver run went from 105 to 34. This ranks as the eleventh highest terminating sequence.

On March 7th, 92898 terminated after having reached a maximum of 107 digits. Paul Zimmermann originally computed this to 80 digits where it was guided by 2^4 * 3 at index 1084. I extended it to index 2459 where it was driven by 2^2 * 7 at 92 digits. Wolfgang Creyaufmüller then advanced it to an escape from that driver, leaving off at 101 digits on August 28, 2005 where it was guided by 2^4 * 3 * 5. Continuing on August 30, 2005, I captured the downdriver at 107 digits on May 2nd, 2006, leaving off with the 2^2 * 7 driver at the height of 102. Encouraged by the recent success Mersenne Forum members were having escaping drivers in the 5-digit range, I began attempting the same on my own sequences that previously had downdriver runs beginning in triple digits. For 92898, that process began March 6th. Escape from 2^2 * 7 came after only 11 steps, with the downdriver soon recaptured, leading to termination the next day.

On November 18, 2009, 98616 terminated after reaching a maximum of 118 digits: 98616i2790=43. Paul Zimmermann originally computed this sequence to the target height of 80 digits where it was driven by 2 * 3^2. In April of 2003, Wolfgang Creyaufmüller escaped this driver at 90 digits, leaving off at the size of 99 where it was then driven by 2^3 * 3^2 * 5. In August of 2005, he added 14 more lines, increasing the size to 104. On February 26, 2008 I began working from there. After reaching 118 digits, the downdriver was obtained at 115, reducing the size to 99. The downdriver was acquired a second time at 115 digits, this time leading to a finish.

On September 3rd, 2008, 62850 broke the record for the highest terminating sequence one week after aquiring the downdriver at 121 digits. Paul Zimmermann originally computed this sequence to the target height of 80 digits where it was guided by 2^5 * 3 at index 740. I continued from there, and obtained the downdriver at 90 digits, resulting in a drop to the height of only 18. Eventually it rebounded to 94 digits, where I left off at index 2948 in March of 2003. Wolfgang Creyaufmüller advanced the sequence to 101 digits in February 2005, where it was driven by 2^3 * 3^2 at index 2978. I continued computations on July 15, 2006, with the downdriver obtained on August 27th of this year. That produced the second longest downdriver run, reducing the size by 82 digits. It finished at index 3973 in the prime 41.

On July 20th, 2008, 98790 terminated after reaching a height of 112: 98790i4443=13. Paul Zimmermann originally computed this sequence to the target height of 80 at index 3115 where it was guided by 2^5 * 3. I extended it to 99 digits where it was driven by 2^3 * 3^3 * 5 at index 3200 on October 2002. Wolfgang Creyaufmüller moved it past the 100-digit mark at index 3209 and size 103 in August of 2005. I restarted computations from there on July 7th, with the downdriver acquired on July 17th, leading to termination three days later.

In the evening of September 4, 2007, 11670 terminated after reaching a new record height (for terminating sequences) of 113. Wieb Bosma originally computed this sequence to 91 digits where it was guided by 2^6 * 3 * 5 at index 954. Wolfgang Creyaufmüller advanced it to a height of 101 at index 979 on November 2, 2003 where it was at the last line of a run by 2^5 * 3^2. I began working from there on December 25, 2005, eventually leaving off at 1024. 2^2 * 3 * C109 sz 110 on August 30, 2006. The addition of GGNFS to my factoring resources on May 6th of 2007 enabled the resumption of work on this sequence on August 23rd, and on that date it achieved stability for the first time in this century. The downdriver was acquired on August 30th at height 113, which eventually led to termination at index 3534 with the prime 193.

On May 5, 2007, 15960 terminated after reaching a new record height (for terminating sequences) of 109. Wieb Bosma originally computed this sequence to the target height of 90 digits at index 749, where it was driven by 2^3 * 3^3 * 5. Wolfgang Creyaufmüller extended it to 100 digits in November of 2003, achieving an escape from from that driver and leaving off at index 782 where it was guided by 2^2 * 3. On February 20, 2005 I began from there, working off and on over the next couple of years. Eventually, the downdriver was obtained at height 106 on May 3rd, and it terminated two days later at index 2033 and prime 41.

On September 16, 2006, 77880 was found to be a side-sequence of 21546: 77880:i1470 = 21546:i72 = 1373808. Paul Zimmermann originally computed this sequence to the target height of 80 at index 454 where it was being driven by 2^3 * 3 * 5. As part of a project to advance all sequences in [50,000, 100,000] to a height of at least 90, Wolfgang Creyaufmüller brought 77880 to a height of 93 at index 494 on May 27, 2003 where it was then driven by 2^2 * 7. Two years later, Wolfgang advanced the sequence to the new target height of 100 at index 541. His computations included an escape from 2^2 * 7 at 98 digits, and finished on May 29, 2005 under the control of 2^3 * 3^2. On September 15th, 2006, I began work to further extend the sequence. It quickly escaped from that driver, and acquired the downdriver at its maximum height of 103. A precipitous decline quickly ensued, culminating in the merger with 21546 the next day. This is a new record for the greatest height reached by a side-sequence (prior to confluence). The previous record was held by 72288 and 109086, both of which reached a maximum of 90 digits.

Near midnight of May 29, 2006, 45984 ended in the prime 11 at index 1490 after reaching a new record height (for terminating sequences) of 107. Wieb Bosma originally computed this sequence to the target height of 90 digits, at index 499. My initial efforts reached an impasse on March 15, 2002 at index 589 and height 101. I added a few lines on October 15, 2004, leaving off at height 103 and index 598, stuck on a 100-digit composite. Things finally got going again on December 18, 2005. After reaching a maximum height of 107 digits at index 841, the downdriver was obtained on May 27, 2006 at 106 digits, which produced an exceptionally long drop to 54 digits. Only brief interruptions separated the succeeding downdriver runs which led to a successful conclusion.

17130 ended in the perfect number 8128 at index 3225 on August 26, 2005. Wieb Bosma originally computed this sequence to 98 digits at index 1498, where it was guided by 2^4 * 3^2. In December of 2003, Wolfgang Creyaufmüller continued computations, finding that it quickly lost the 3 and acquired the downdriver at height 99 which resulted in a decline to 80 digits. Eventually it came under control of 2^3 * 5, and Wolfgang left off at the target height of 100 digits at index 1827. On March 9th of 2005 I added a single line, and did not do anything more until August 23rd of that year. The sequence escaped from 2^3 * 5 at index 1836 and acquired the downdriver at height 101, which quickly led to its termination.

21024 terminated on April 30, 2005 after reaching a new record height (for terminating sequences) of 103 digits: 21024i2423=1429. Wieb Bosma originally computed 21024 to 98 digits at index 1011. In January of 2004, Wolfgang Creyaufmüller extended it to 101 digits at index 1023, leaving off at 2^4 * 3^2 * 131 * C97. On April 14th of 2005, I began working where Wolfgang left off. After reaching 103 digits, the downdriver was obtained on April 18th at a height of 102 digits, with termination achieved twelve days later.

56368 finished on April 5, 2005, breaking the record set by 3630 for terminating sequences by reaching 102 digits: 56368i3446=43. Paul Zimmermann initially computed this to a height of 80 digits at index 789 where it was under control of the 2^6 * 127 driver. Wolfgang Creyaufmüller extended it all the way to index 1931 after observing a quick escape from that driver at the 81-digit mark and a big drop to 36 digits. He left off on July 11, 2003 at the target height of 90 where it had become stable at 2^2 * 5. Two days later my efforts began, resulting in an advance to 100 digits on December 16, 2003 at index 1992 where it stalled at 2^2 * 3 * 41 * 53 * C95. After a gap of more than a year, I resumed work with a faster computer on January 7, 2005. After reaching 102 digits, the downdriver was obtained on March 31 which led to a disappointing drop of only 100 to 95 digits. A brief interlude with 2^4 * 31 was followed by another aquisition of the downdriver which was much more productive, culminating in the prime 43 being reached the next day.

62832:i1740 = 43 on June 28, 2002, reached a maximum of 100 digits. My computations began where Paul Zimmermann left off at 80 digits.

23910 terminated on April 21, 2004 after reaching a maximum of 100 digits by entering the amicable pair (1210, 1184): 23910i1891=1210. Computations began where Wolfgang Creyaufmüller left off at 100 digits. Wieb Bosma had originally computed this to 92 digits at index 1121.

105384 terminated on April 1, 2004 after reaching a maximum of 96 digits by entering the same amicable pair: 105384i2846=1210. This time my computations began where Juan L. Varona and Manuel Benito had left off at 80 digits.

72288 reached a height of 90 digits before merging with 11408 on July 17th, 2002: 72288:i1642 = 11408:i6 = 85808. Paul Zimmermann had initially computed this to 80 digits.

Could the Catalan-Dickson Conjecture be true?

Empirical evidence and mathematical analysis seems to suggest that there are infinitely many open-ended sequences. However, there is disagreement among mathematicians on this issue. Consider that sometimes when a sequence aquires the downdriver, the result is a precipitous drop to termination with only small interruptions along the way. It might be assumed that every sequence encounters the downdriver at various points. Perhaps it's just a matter of reaching one favorable to quick termination. In practice, aquisition of the downdriver after a sequence reaches 80 digits rarely initiates a sustained drive to the bottom. When that driver is lost, the sequence is once again subject to the normal upward forces. To actually make it back to an endpoint, several additional lucky runs are required.

There has been a dramatic reduction in the number of new terminating sequences as we moved from heights in the 80s, 90s, and now to the 100s. It took nearly four years after Juan L. Varona and Manuel Benito terminated 3630 before a new record was achieved (56368), and the former continues to be the highest a terminating sequence has reached in the select region below 10^4. In spite of our access to faster computers and improved software, only nine sequences have terminated after reaching a height greater than 110 digits. The long obstacle course that must be negotiated from such levels has proved daunting indeed.

Unresolved sequences in [1, 10^4] that began a downdriver run after reaching 100 digits

 n maximum height* downdriver range ultimate bottom computed by downdriver found by 564 104 104-83 62 Lehmer/ Lenstra/ Montgomery/ Wagstaff/ Creyaufmüller/ Zimmermann Zimmermann 660 145 145-138 138 Lehmer/ Lenstra/ Montgomery/ Wagstaff/ Creyaufmüller/ Zimmermann Zimmermann 1488 107 107-106 106 Varona/ Kleinjung Kleinjung 1578 111 110-57 106-96 5 93 Varona/ Clavier/ Creyaufmüller/ Hoogendoorn Clavier Hoogendoorn 1578 132 132-95 103-72 112-98 111-100 103-78 91 47 98 100 54 Clavier Clavier 1734 111 111-103 103-71 103   70 Varona/ Clavier/ Schein Schein 2232 113 112-108 108 Varona/ Clavier Clavier 2712 123 123-102 102 Varona/ Clavier Clavier 3366 106 106-100 103-86 99 85 Varona/ Clavier Clavier 3564 115 115-73 103-81 73 37 Varona/ Clavier/ Johnson Johnson 3678 111 111-68 23 Varona/ Hudson/ Clavier Hudson 3906 158 158-131 131 Varona/ Clavier/ Leclair Leclair 4116 104 104-93 92 Varona/ Benito Varona/ Benito 4380 124 124-100 103-78 78 Varona/ Clavier Clavier 6822 137 131 129-41 131-63 32 59 Varona/ Clavier Clavier 7560 100 100-81 37 Varona/ Benito Varona/ Benito 7890 123 116-102 102 Varona/ Clavier/ Schein Schein 8760 102 99-51 92-36 28 20 Varona/ Benito Varona/ Benito 9462 129 129-116 116-107 105-91   91-84 84 Varona/ Leclair Leclair 9684 109 110 108-96 109-107 96 107 Varona/ Clavier Clavier 9684 134 133-113 111-88 39 Varona/ Clavier/ Chevallier-Mames Chevallier-Mames 9852 131 129-92 90 Varona/ Clavier/ Hoogendoorn Hoogendoorn

* This column refers to the maximum height each sequence had attained before initiation of a downdriver run.

A downdriver run is defined here to be a series of at least two consecutive iterations containing the downdriver, and is considered to have ended upon reaching two consecutive lines that both do not have the downdriver.

Note: The table for sequences beginning with five-digit numbers has grown too large, and has therefore been moved here.

Utilities for aliquot sequences

The programs in routines.zip perform various functions:

renumber.c adds lines from one file to another if a confluence exists between them. The added lines are renumbered.
num2elf.c converts the "numbers" type files of Ivo Duentsch to the .elf format.
elf2num.c is the reverse of num2elf.
cl2elf.c converts the files of Christophe Clavier to the .elf type.
elf2cl.c is the reverse of cl2elf.
merge.ub locates the first point of confluence between two files that have merged.
ck.ub runs a quick check of the soundness of a file.
check.ub does a thorough check of a file.
uu.bat is a routine for running UBASIC programs from the command line.
seq.c writes the last portion of an aliquot file to seq.txt. The number of each line is removed and replaced by its size, which is given at the end of the line. The purpose is to make it easier to follow the course of a sequence beyond a point specified by the user. Enter the sequence number as the first argument, followed by the starting line number as the second.

32-bit DOS executables: renumber.exe  cl2elf.exe  elf2cl.exe  num2elf.exe  elf2num.exe  seq.exe

Here is a second set of programs, contained in set2.zip

chk.c tests the soundness of an aliquot file, checking for each of the possible errors. Enter the sequence number and one of the standard files will be opened: elf, num or those of Christophe Clavier. Alternatively, enter the complete file name as the argument. If an error is is found, the details are displayed and the program exits. Therefore, it should be run again after repairing the faulty line to check for additional errors. This function can be utilized in a program by testing for an error code of 1 to determine if a file is unsound. It can thus be employed to test multiple files automatically.

alk.c is a new aliquot sequence compression function similar to that of Jesper Gerved, used extensively on this web site. Unfortunately, the latter is limited to sequences that reach a maximum of 144 digits, a level that has been frequently surpassed of late. Enter an argument in the same manner as chk.c: if only the sequence number is given, the program will be applied to one of the standard types, searching in the order .elf, Clavier then num. Alternatively, enter the full name of any file whose contents are of the standard type. A check is first made of the file's soundness, as the process will not succeed if any errors are present. After passing that test, the essential data is saved in a file with the extension .alk. To decompress, enter just the sequence number, and the .alk file will be detected with the original contents reconstructed as an .elf file. When both an .elf and .alk file exist for a given sequence and only the sequence number is entered as the argument, the situation is ambiguous. Here is how it's handled: a comparison is made between the last index of the two files. If the index of the .elf file is bigger, then it is assumed that compression is desired, and a new .alk file is created. This prevents lines added since the last compression from being wiped out by the outdated .alk file. In either case, the result is that both files are up to date. Though .alk files are bigger than the corresponding .alq's, the new ones become much smaller after zip compression is applied compared to zipped alq's.

errfix.c repairs formatting errors in the files of Christophe Clavier. These include repeated factors and the lack of a space between an asterisk and a number. Repeated factors are combined into one along with the correct exponent, and a space is inserted between an asterisk and number where necessary. Enter just the sequence number, and the corresponding text file will be opened. It might be necessary to run this routine before applying chk.c to one of these files in order for it to pass the test. This program has been newly revised to handle 28 lines recently added to 3366 which contain no spaces at all between factors and asterisks.

quadratc.c applies the quadratic formula to finish the factorization of a number when the factors have been lost but the next number of the aliquot sequence is known. If the factorization cannot be readily determined by normal means, it could be because the last two factors are large and a lengthy factoring job might be necessary. At this point, the quadratic formula will finish the task. To begin, execute the function and enter numbers for n1 and n2 when prompted: the first is the original number to be factored and the second is the next number in the sequence. Alternatively, place the two numbers in a file and enter the file name as the function's argument. The program trial divides by the primes less than 2^17 and then tries to finish it off with the quadratic formula. If that fails, it means that more than two factors remain, and the user is prompted to supply an intermediate factor. This continues until the factorization is completed.

mgma2elf.c (magma to elf) converts the files of Wieb Bosma (which are in Magma format) to the standard .elf type. Enter the sequence number and the program attempts to open the corresponding file using the naming scheme Bosma employs. So far, only two of these files have been hyperlinked (see this page), but when others are available this function will be useful.

appendat.c (append .dat file) facilitates distributed computing for Msieve's quadratic sieve. It appends the contents of the .dat file found in a given location (whose path is entered as the function's argument) to the one present in the current directory. A check is made to verify the corresponding composites are the same. For Yafu, use appendyf.c (see below).

convert.pl is Perl script that converts the output of Msieve's polynomial selection to the form used by GGNFS. This was written according to the detailed instructions gratefully provided by Jeff Gilchrist. The script calls for the project name as the argument. Therefore, to run GGNFS with Msieve creating the .poly file, use the following steps:

msieve -v -np -i <name>.n
convert.pl <name>
factmsieve.pl <name>.n

32-bit DOS executables: chk.exe  alk.exe  errfix.exe  quadratc.exe  mgma2elf.exe  appendat.exe  appendyf.exe

If the second core of a computer is available and not in use, then the speed of Yafu can be doubled. After beginning execution of that function, switch to another directory and execute Yafu again for the same composite. When the sum of the two relation sets reaches somewhat more than half of the required number, stop both, execute appendyf in the first directory using the address of the second as the argument and restart Yafu. If significantly more relations remain to be accumulated, go to the second directory, delete siqs.dat and run Yafu again there. After a little while, stop both and repeat as explained above. The same process can be used for Msieve, using appendat instead of appendyf. For two Yafu's, try the combining process when both reach 17,000 relations. For Msieve, look for the relation sums to reach 45% to 50% of the required number before switching to the combining phase.

Averages for downdriver runs
(These are based on the 2494 sequences below 251712)

 starting size number of lines digit reduction 20-29 24.16 4.91 30-39 34.61 7.08 40-49 44.63 9.04 50-59 58.30 11.81 60-69 66.02 13.45 70-79 83.40 16.94 80-89 88.47 17.91 90-99 98.90 20.09 100+ 116.75 23.67

Record downdriver runs

 length sequence index 565 139314 1724 536 162126 2057 471 707016 5265 436 6822 1595 406 62850 3228 394 921232 5753 388 149808 1062 386 33960 1473 374 740106 2863 369 37968 553 363 157752 1226 362 80232 1095 355 574614 1197 353 15390 799 345 743370 723 343 781302 825 342 570690 1295 341 722064 543 341 753732 1847 336 233324 2454 336 513300 346 335 17892 973 335 561000 744 333 6822 3016

Record reductions in size from a downdriver run:

 reduction sequence index 108 139314 1724 101 162126 2057 90 707016 5265 88 6822 1595 82 62850 3228 81 33960 1473 81 37968 553 80 921232 5753 78 149808 1062 75 80232 1095 75 740106 2863 75 829914 2016 73 29052 1347 72 157752 1226 70 722064 543 69 743370 724 69 753732 1847 68 6822 3016 68 29070 1075 68 561000 744 67 147324 595 67 199152 2450 66 154056 1345 66 574614 1197

Highest downdriver aquisitions in [1, 10^4]

 height sequence index obtained by date 158 3906 1571 Don Leclair May 28, 2010 145 660 728 Paul Zimmermann July, 2006 142 1560 1650 Christophe Clavier September 29, 2009 133 9684 1234 Benoit Chevallier-Mames January 12, 2007 132 1578 4359 Christophe Clavier April 6, 2006 131 6822 3016 Christophe Clavier June, 2009 129 6822 1595 Christophe Clavier June, 2008 129 9462 552 Don Leclair August, 2005 129 9852 1003 Sander Hoogendoorn Summer, 2008 124 4380 1182 Christophe Clavier October 5, 2009 123 2712 1525 Christophe Clavier December 23, 2005 116 7890 1360 Igor Schein February, 2006 116 9462 798 Don Leclair December, 2005 115 3564 895 Donovan Johnson September, 2005 112 1578 5964 Christophe Clavier May 11, 2006 112 2232 478 Christophe Clavier 2004 111 1578 6278 Christophe Clavier May 27, 2006 111 1734 1484 Igor Schein August, 2005 111 9684 1441 Benoit Chevallier-Mames February, 2007 110 1578 1185 Christophe Clavier February, 2005 110 3678 1396 Mark Hudson March, 2005 110 3906 1250 Don Leclair August, 2005 109 9684 1026 Christophe Clavier October 12, 2005 108 3906 829 Don Leclair August, 2005 108 9684 696 Christophe Clavier 2004 107 1488 919 Thorsten Kleinjung January, 2005

Highest downdriver aquisitions in [10^4, 10^6]

 height sequence index date 14712 11040 6174 September, 2010 1418 199152 4646 May 21, 2009 1408 199152 4587 April 28, 2009 13810 147192 4629 February, 2010 13612 15390 799 March, 2010 1362 170196 1783 February 18, 2010 1343 162126 2057 March 21, 2009 1332 29772 1025 October 18, 2009 1335 103920 4581 April 17, 2009 131 20076 1676 December 14, 2010 1312 171018 1971 April 14, 2010 13110 205980 2296 August, 2010 1315 139314 1724 May, 2009 130 29360 1456 September 5, 2010 1296 113454 2120 May, 2009 12915 157752 4457 2009-2010 1292 604560 2474 December 2, 2009 128 14280 925 October 23, 2010 12816 34308 1461 May 15, 2010 128 76014 1158 June 27, 2010 1282 144984 6650 April 4, 2008 1287 182224 1196 April, 2009 12616 80244 1139 2010 126 76314 1157 May 30, 2010 12612 921232 5640 April, 2010 125 21528 978 April 8, 2010 125 36960 1634 May 3, 2010 12510 205980 2499 August, 2010 124 11408 937 October 17, 2010 124 12120 1225 March 10, 2010 124 81576 713 September 22, 2010 1242 88662 2185 November 4, 2009 1245 128370 3646 April 10, 2009 12316 84822 3081 2010 1239 107310 2521 May, 2009 12312 15390 656 2010 12213 34308 928 2010 122 47352 1465 July 24, 2009 1222 47728 2913 February 26, 2008 122 49692 947 July 7, 2009 122 65208 2310 March 11, 2009 122 68340 769 March 24, 2009 121 25380 1106 January 25, 2009 121 37146 1646 October 28, 2008 121 62850 1984 August 27, 2008 121 77840 3869 December 5, 2008 1212 88662 1796 June 24, 2009 1218 167148 2666 May, 2009 12113 232956 1490 2009 120 14970 1551 June 1, 2008 120 32280 2300 July 19, 2009 120 37146 1288 September 5, 2008 12616 80244 978 2010 12013 832194 1205 2010 119 20076 1093 February 6, 2009 119 52218 1635 August 2, 2009 119 80232 980 September 13, 2008 119 95610 915 November 27, 2009 11910 114810 3194 June, 2009 1197 182224 1544 May, 2009 11912 735966 3065 December, 2009 11816 34308 1117 2010 118 75798 434 July 8, 2008 1182 163716 932 May 27, 2009 11712 11040 5524 October, 2009 117 39870 1554 April 14, 2008 117 62820 849 October 9, 2008 117 70464 684 May 8, 2008 1176 181410 1641 April, 2009 1166 10212 1614 December 31, 2008 116 12120 1052 November 7, 2009 116 18336 2024 October 21, 2009 116 19866 1780 January 21, 2010 116 77840 3083 April 2, 2008 1168 199152 1138 April, 2009 11511 14994 1664 2009-2010 115 45792 847 October 6, 2008 1152 48462 1294 April 1, 2007 115 80232 1095 September 25, 2008 115 82728 2418 November 23, 2007 115 98616 1141 November 7, 2009 115 98616 1426 November 15, 2009 1152 83064 798 March 6, 2008 11511 151752 1209 June, 2009 11515 157752 785 2009 1152 171018 1008 August 25, 2009 1154 175182 1050 March, 2009 11511 507924 690 September 13, 2009 11512 519330 1555 August, 2009 114 12120 1052 November 7, 2009 114 17544 850 December 8, 2007 114 19380 1191 July 11, 2007 114 19866 2409 February 6, 2010 114 20076 1354 February 26, 2009 11413 26820 1944 2010 114 37146 3994 November 22, 2008 114 82236 871 October 2, 2007 1147 505536 483 August, 2009 11410 737160 2145 December, 2009
2Frank Schickel
3Markus Tervooren
4Max Dettweiler
5Mikael Klasson
6Andreas Schindel
7kriZp
8Ignacio Santos
9Greg Childers
10Serge Batalov
11Jonathan DeMarrais
12Dmitry Domanov
13Mersenne Forum
14Karsten Bonath
15David Stevens
16Tom Womack

Clifford Stern
Santa Monica, California
ax810@lafn.org