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Topic: Aliquot sequence

	Aliquot sequences (with leading terme < 10,000)
		Note that the seven sequences 276, 552, 564, 660, 966, 1074 and 1134 are pursued by Paul Zimmermann.
		Sequence 56,440 which was unfinished can be removed from tables as it is now linked to sequence 1578.
		Sequence reached a maximum at index 1398 with 111 digits, and decreased down to 23 digits at index 1868 on March 15, 2005.
	christophe.clavier.free.fr /Aliquot/site/Aliquot.html (773 words)

	PlanetMath: examples of aliquot sequences
		The shortest aliquot sequence with distinct elements is that for 1, simply: 1, 0.
		Technically, all aliquot sequences are infinite, but some cease to be interesting sooner than others.
		This is version 3 of examples of aliquot sequences, born on 2006-08-09, modified 2006-09-08.
	planetmath.org /encyclopedia/ExamplesOfAliquotSequences.html (268 words)

	Aliquot sequence - Wikipedia, the free encyclopedia
		For instance, the aliquot sequence of 220 is 220, 284, 220, 284,....
		For instance, the aliquot sequence of 95 is 95, 25, 6, 6, 6,....
		Catalan's Aliquot Sequence Conjecture from MathWorld (E.W. Weisstein)
	en.wikipedia.org /wiki/Aliquot_sequence (384 words)

	World's Longest Unitary Aliquot Sequence
		On 11 July 2001, the longest known terminating unitary aliquot sequence was computed to its termination.
		A unitary aliquot sequence is constructed similarly to a "standard" aliquot sequence, except that instead of adding divisors, one adds up unitary divisors.
		It is conjectured that all unitary aliquot sequences eventually either enter a closed cycle, or terminate by reaching the number 1.
	www.brennen.net /primes/151244562.html (420 words)

	CATHOLIC ENCYCLOPEDIA: Prose or Sequence
		A sequence usually begins with an independent introductory sentence or an Alleluia (an intonation with its own melody); then follow several pairs of strophes, each pair with its own melody; in the earlier periods the conclusion is uniformly an independent sentence of shorter or longer form.
		Sequences with more or less numerous traces of the transition process are so numerous that they may be placed in a class by themselves.
		In French Graduals almost all the sequences of the first epoch were supplanted by the later ones, whereas in Germany, together with the new ones a considerable number of those which are supposed to be Notker's remained in use as late as the fifteenth century.
	www.newadvent.org /cathen/12481d.htm (4768 words)

	Aliquot Sequences
		The aliquot divisors of a number are all of its divisors except itself.
		An aliquot sequence is a sequence of numbers where each number is the sum of the previous number’s aliquot divisors.
		The ultimate fate of the sequences for all numbers up to 1000 has been found to be one of the first three cases, except for 276, 552, 564, 660, and 966.
	home.netcom.com /~jrhowell/math/aliquot.htm (562 words)

	aliquot part
		For instance, the aliquot parts of 12 are 1, 2, 3, 4, and 6.
		An aliquot sequence is formed by taking the sum of the aliquot parts of a number, adding them to form a new number, then repeating this process on the next number and so on.
		In other cases, where a sequence repeats a pattern after more than one step, the result is known as an aliquot cycle or a sociable chain.
	www.daviddarling.info /encyclopedia/A/aliquot_part.html (263 words)

	Aliquot - Wikipedia, the free encyclopedia
		In mathematics, an aliquot part (or simply aliquot) of an integer is any of its integer divisors.
		For instance, 2 is an aliquot of 12.
		The sum of all the aliquots of an integer n is the value σ(n) of the divisor function σ at n.
	en.wikipedia.org /wiki/Aliquot (103 words)

	Aliquot Sequences from the Trenches
		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.
		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.
		Seventy-six sequences have completed a downdriver run after reaching 100 digits, seven of which terminated and one was resolved as a side-sequence.
	www.lafn.org /~ax810/aliquot.htm (2964 words)

	[No title]
		The sequence obtained by iterating s for some n is called an aliquot sequence or, sometimes, a chain.
		The sequence is constant from some point onwards; the constant term will be 1 or a perfect number.
		The target of an aliquot sequence is a member which is either a perfect number or a prime or the smallest member of an amicable ring.
	archives.math.utk.edu /software/msdos/dynamics/aliquot/aliquot2.readme (822 words)

	PlanetMath: aliquot sequence
		is an amicable number, its aliquot sequence is periodic, alternating between the abundant and deficient member of the amicable pair.
		In other cases, the aliquot sequence reaches a
		This is version 2 of aliquot sequence, born on 2006-07-28, modified 2006-08-09.
	planetmath.org /encyclopedia/AliquotSequence.html (87 words)

	Aliquot sequences (Site not responding. Last check: )
		Aliquot sequences arise in iterating the sum-of-divisors function s, which assigns to a positive integer the sum of its proper divisors (i.e., excluding the number itself).
		For all n up to 50000 I have computed terms of the aliquot sequence starting with n until it terminated, entered an aliquot cycle, or reached an entry with at least 80 decimal digits.
		All this is ongoing work as part of a bigger effort by several people (see the aliquot page of Wolfgang Creyaufmüller, or the page of Juan Varona, or the aliquot page of Paul Zimmermann).
	www.math.ru.nl /~bosma/research/nuth/ali.html (404 words)

	Problem 22.- Divisors (IV). Aliquot Sequences. Factorizing s1131 (276).
		(n),….} is a sequence of numbers where each member is the sum of all the divisors of the previous member, excluding the member itself.
		Lenstra has proved that it is possible to construct arbitrarily long monotonic increasing aliquot sequences…" Paul Zimmerman believes different: " My personal opinion is that Catalan's conjecture is true, that is every aliquot sequence converges to a prime, or a perfect number, or a cycle of amicable or sociable numbers...
		Thanks to a kind e-mail of Paul Zimmerman (31/07/99) the current status of the aliquot sequence for n = 276 goes up the index k = 1131, which is composite of 108 digits which contains a composite number of 97 digits still pending to be factorized.
	www.primepuzzles.net /problems/prob_022.htm (762 words)

	Global Peptide Services, LLC: FAQ
		Many times it is unclear what sequence compositions will be troublesome during synthesis, and while some sequences are relatively easy to synthesize they present unexpected difficulties during purification.
		Ideally, peptide sequences should be 10-30 residues in length, be relatively hydrophilic and possess few or none of the avoidances discussed in Question 2.
		It must be remembered that in addition to avoiding difficult sequence compositions, long sequences contribute to a peptide’s difficulty and achieved purity.
	www.globalpeptide.com /faq.html (1357 words)

	Perfect numbers
		Today the usual definition of a perfect number is in terms of its divisors, but early definitions were in terms of the 'aliquot parts' of a number.
		An aliquot part of a number is a proper quotient of the number.
		Note that 10 is not an aliquot part of 10 since it is not a proper quotient, i.e.
	www-groups.dcs.st-and.ac.uk /~history/HistTopics/Perfect_numbers.html (4360 words)

	Index to OEIS (Section Al)
		aliquot sequence (or trajectory) for n, length of: A098007 , A098008 , A003023, A044050 *, A007906, A003062
		aliquot trajectories for certain initial values: (1) A008885 A008886 A008887 A008888 A008889 A008890 A008891 A008892 A014360 A014361 A074907 A014362
		aliquot trajectories for certain initial values: (2) A045477 A014363 A014364 A014365 A074906
	www.research.att.com /~njas/sequences/Sindx_Al.html (217 words)

	ALIQUOT PAGE - Prime families - Primzahlfamilien
		An aliquot sequence is a sequence of integers, built with the sigma function.
		An open-end sequence with all side sequences is called an open-end family (OE-family).
		ALIQUOT is written by Ivo Duentsch in Turbo Pascal and available in its original version at the ftp-server.
	www.aliquot.de /aliquote.htm (3347 words)

	Duke Bryology Laboratory - Lab Protocols
		Aliquot appropriate quantity of master mix (24.7µl or 49.5µl) to clean, labeled PCR tubes held in a rack on ice.
		Aliquot 5.0µl of master mix into the tubes in the appropriate column (refer to grid diagram).
		For a region where the sequence is not complete (such as for the 26S, where the 3' end may be incomplete), click "3' (or 5') partial".
	www.biology.duke.edu /bryology/protocols.html (5455 words)

	Aliquot sequences starting under 10000
		In this table we summarize the present state of aliquot sequences starting with under 10000.
		we mean that all the other sequences with unknown end merges with any sequence in the table; in this cases, we only show in the table the one that starts in the smaller number).
		Wolfgang Creyaufmueller and Paul Zimmermann for the sequences starting in the numbers 276, 552, 564, 660, 966, 1074 and 1134 (these values can be out of date in this table).
	www.unirioja.es /dptos/dmc/jvarona/table.html (180 words)

	Math Is Fun Forum / Prime Numbers - Advanced
		Aliquot sequences - every number has its own Aliquot sequence, and lots of Aliquot sequences are the same - they just start in different places.
		An Aliquot sequence terminates when it reaches a number already found (forming a sociable group) or when 1 is reached, because the next number will automatically be zero.
		An amicable number's aliquot sequence consists of itself and its amicable partner...
	www.mathsisfun.com /forum/viewtopic.php?pid=35044 (486 words)

	Problem 23.- Divisors (V) Aliquot sequences, Sociable Numbers.
		As we saw in the Problem 22, one possible end of an aliquot sequence is to reach to a cycle of numbers.
		Moews and I have exhaustively searched for all aliquot cycles of length greater than 2 such that the number preceding the largest number of the cycle is below 1.54*10^11.
		I have verified the existence of some of these cycles, and found that the least number that generates - in some instant of the develop of the aliquot sequence - the cycle of order k = 5 is n = 9464.
	www.primepuzzles.net /problems/prob_023.htm (480 words)

	Scanning the human proteome for calmodulin-binding proteins -- Shen et al. 102 (17): 5969 Data Supplement - HTML Page - ... (Site not responding. Last check: )
		An aliquot of the TNT mixture was incubated with an appropriate amount of biotinylated CaM in the presence of 1 mM CaCl
		Aliquots of the TNT expressed protein (C), the last wash (W), and the eluent (E) were loaded onto an SDS/PAGE gel for binding analysis.
		The complete sequence information for all fragments isolated from each gene and the mapped shortest overlap region is available upon request.
	www.pnas.org /cgi/content/full/0407928102/DC1 (1461 words)

	Photocross-linking of the Homing Endonuclease PI-SceI to Its Recognition Sequence -- Pingoud et al. 274 (15): 10235 -- ...
		Double-stranded oligodeoxynucleotides comprising either the full-length PI-SceI recognition sequence (in capital letters) or the right-half cleavage product were monosubstituted with 5-IdU in the cases of T, A, or G or 5-IdC in the case of C. The positions (numbered as proposed by Gimble and Wang (6)) that were substituted are indicated by circles or arrows.
		Aliquots of the fractions 28-35 were precipitated and analyzed
		Aliquots before (lanes 1 and 3) and after irradiation (lanes 2 and 4) were analyzed by electrophoresis on a 12% (w/v) SDS-polyacrylamide gel that was silver-stained.
	www.jbc.org /cgi/content/full/274/15/10235 (6916 words)

	Yeast Barcode Microarrays
		At various times, aliquots of cells were removed and the genomic DNA was extracted and amplified using the same two PCR primers.
		Genomic DNA from an aliquot of this initial population of cells was used as template for PCR using the two primer sequences shared by each of the deletion strains.
		After a period of time, an aliquot of cells was again used for template of PCR using the same two PCR primers.
	www.bio.davidson.edu /courses/genomics/method/barcode.html (687 words)

	A protein subunit of human RNase P, Rpp14, and its interacting partner, OIP2, have 3'right-arrow5' exoribonuclease ...
		Aliquots of each immunoprecipitate and substrate concentration used in each reaction are given in the legend of Fig.
		Aliquots for each immunoprecipitate and substrate concentration are given in the legend to Fig.
		Aliquots as defined earlier, as was substrate concentration (Fig.
	www.pnas.org /cgi/content/full/99/8/5295 (3773 words)

	Hartwell Center/Macromolecular Synthesis/Nucleic Acid Synthesis
		Orders delivered this way can be released significantly faster than any other option, and an aliquot of several micro liters can be dried down for quantification and use in just a few minutes in your lab.
		When entering the sequences, substitute T for U. If you require dT at the 3’ terminal bases, please specify this in the comment box on the order form.
		Working aliquots may be stored at 4°C for up to one month.
	www.hartwellcenter.org /bio_services/macsyn/DNAservices/index.php (1415 words)

	Aliquot sequence 3630 ends after reaching 100 digits, Manuel Benito, Wolfgang Creyaufmüller, Juan L. Varona, Paul ...
		In this paper we present a new computational record: the aliquot sequence starting at 3630 converges to 1 after reaching a hundred decimal digits.
		Also, we show the current status of all the aliquot sequences starting with a number smaller than 10,000; we have reached at least 95 digits for all of them.
		in particular, we have reached at least 112 digits for the so-called "Lehmer five sequences," and 101 digits for the "Godwin twelve sequences." Finally, we give a summar showing the number of aliquot sequences of unknown end starting with a number less than or equal 10^6.
	projecteuclid.org /getRecord?id=euclid.em/1062621215 (240 words)

	Neglected Gaussians
		The numbers 220 and 284 are amicable, because each is the sum of the divisors of the other.
		aliquot sequence can terminate with 0, grow without bound to infinity, or go into a cycle.
		One aliquot cycle of length 5, found in 1918 by Poulet, is 12496, 14288, 15472, 14536, and 14264.
	www.mathpuzzle.com /Gaussians.html (723 words)

	LUMINESCENCE DATING OF GLACIAL LAKE GREAT FALLS, MONTANA, U.S.A
		Optically-stimulated luminescence (OSL), using the single aliquot SAR method, gave an age of 17.1±1.4 ka for the lake silts.
		A stratigraphic sequence near Hower Coulee, Montana, contains fluvial gravels and sands that are buried by silts (the “lower lake”) of glacial Lake Great Falls.
		There are two ages on sandy and silty lithofacies of the upper lake sequence, using the single-grain leading edge technique: 10.5±3.33 ka (UW454) and 14.3±3.66 ka (UW467).
	gsa.confex.com /gsa/inqu/finalprogram/abstract_53968.htm (530 words)

	Richard Guy \| Mathematics and Statistics
		Sequences associated with a problem of {T}ur\'an and other problems North-Holland (1970): 553-569
		Guy, Richard and Selfridge, J. Corrigendum to: ``{W}hat drives an aliquot sequence?'' [{M}ath.
		Guy, Richard and Williams, M. Aliquot sequences near {$10\sp{12}$} (1975): 387-406
	www.math.ucalgary.ca /general/directory.php3?personid=36&capacityid=22 (1251 words)

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