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- Aliquot_sequence abstract "In mathematics, an aliquot sequence is a recursive sequence in which each term is the sum of the proper divisors of the previous term. The aliquot sequence starting with a positive integer k can be defined formally in terms of the sum-of-divisors function σ1 in the following way: s0 = k sn = σ1(sn−1) − sn−1.For example, the aliquot sequence of 10 is 10, 8, 7, 1, 0 because:σ1(10) − 10 = 5 + 2 + 1 = 8σ1(8) − 8 = 4 + 2 + 1 = 7σ1(7) − 7 = 1σ1(1) − 1 = 0Many aliquot sequences terminate at zero (sequence A080907 in OEIS); all such sequences necessarily end with a prime number followed by 1 (since the only proper divisor of a prime is 1), followed by 0 (since 1 has no proper divisors). There are a variety of ways in which an aliquot sequence might not terminate: A perfect number has a repeating aliquot sequence of period 1. The aliquot sequence of 6, for example, is 6, 6, 6, 6, ... An amicable number has a repeating aliquot sequence of period 2. For instance, the aliquot sequence of 220 is 220, 284, 220, 284, ... A sociable number has a repeating aliquot sequence of period 3 or greater. (Sometimes the term sociable number is used to encompass amicable numbers as well.) For instance, the aliquot sequence of 1264460 is 1264460, 1547860, 1727636, 1305184, 1264460, ... Some numbers have an aliquot sequence which is eventually periodic, but the number itself is not perfect, amicable, or sociable. For instance, the aliquot sequence of 95 is 95, 25, 6, 6, 6, 6, ... . Numbers like 95 that are not perfect, but have an eventually repeating aliquot sequence of period 1 are called aspiring numbers (OEIS A063769).The lengths of the Aliquot sequences that start at n are1, 2, 2, 3, 2, 1, 2, 3, 4, 4, 2, 7, 2, 5, 5, 6, 2, 4, 2, 7, 3, 6, 2, 5, 1, 7, 3, 1, 2, 15, 2, 3, 6, 8, 3, 4, 2, 7, 3, 4, 2, 14, 2, 5, 7, 8, 2, 6, 4, 3, ... (sequence A044050 in OEIS)The final terms (excluding 1) of the Aliquot sequences that start at n are1, 2, 3, 3, 5, 6, 7, 7, 3, 7, 11, 3, 13, 7, 3, 3, 17, 11, 19, 7, 11, 7, 23, 17, 6, 3, 13, 28, 29, 3, 31, 31, 3, 7, 13, 17, 37, 7, 17, 43, 41, 3, 43, 43, 3, 3, 47, 41, 7, 43, ... (sequence A115350 in OEIS)Numbers whose Aliquot sequence terminates in 1 are1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, ... (sequence A080907 in OEIS)Numbers whose Aliquot sequence terminates in a perfect number are25, 95, 119, 143, 417, 445, 565, 608, 650, 652, 675, 685, 783, 790, 909, 913, ... (sequence A063769 in OEIS)Numbers whose Aliquot sequence terminates in a cycle with length at least 2 are220, 284, 562, 1064, 1184, 1188, 1210, 1308, 1336, 1380, 1420, 1490, 1604, 1690, 1692, 1772, 1816, 1898, 2008, 2122, 2152, 2172, 2362, ... (sequence A121507 in OEIS)Numbers whose Aliquot sequence is not known to be finite or eventually periodic are276, 306, 396, 552, 564, 660, 696, 780, 828, 888, 966, 996, 1074, 1086, 1098, 1104, 1134, 1218, 1302, 1314, 1320, 1338, 1350, 1356, 1392, 1398, 1410, 1464, 1476, 1488, ... (sequence A131884 in OEIS)An important conjecture due to Catalan with respect to aliquot sequences is that every aliquot sequence ends in one of the above ways–with a prime number, a perfect number, or a set of amicable or sociable numbers. The alternative would be that a number exists whose aliquot sequence is infinite, yet aperiodic. Any one of the many numbers whose aliquot sequences have not been fully determined might be such a number. The first five candidate numbers are called the Lehmer five (named after Dick Lehmer): 276, 552, 564, 660, and 966.As of April 2015, there were 898 positive integers less than 100,000 whose aliquot sequences have not been fully determined, and 9190 such integers less than 1,000,000.".
- Aliquot_sequence wikiPageExternalLink amicable.homepage.dk.
- Aliquot_sequence wikiPageExternalLink Aliquot.html.
- Aliquot_sequence wikiPageExternalLink aliquote.htm.
- Aliquot_sequence wikiPageExternalLink www.aliquotes.com.
- Aliquot_sequence wikiPageExternalLink 3630finishes1.pdf.
- Aliquot_sequence wikiPageExternalLink forumdisplay.php?f=90.
- Aliquot_sequence wikiPageExternalLink Aliquot000.htm.
- Aliquot_sequence wikiPageID "486266".
- Aliquot_sequence wikiPageLength "5966".
- Aliquot_sequence wikiPageOutDegree "16".
- Aliquot_sequence wikiPageRevisionID "666998490".
- Aliquot_sequence wikiPageWikiLink 276_(number).
- Aliquot_sequence wikiPageWikiLink Amicable_number.
- Aliquot_sequence wikiPageWikiLink Amicable_numbers.
- Aliquot_sequence wikiPageWikiLink Aperiodic.
- Aliquot_sequence wikiPageWikiLink Category:Arithmetic_functions.
- Aliquot_sequence wikiPageWikiLink Category:Divisor_function.
- Aliquot_sequence wikiPageWikiLink Conjecture.
- Aliquot_sequence wikiPageWikiLink Derrick_Henry_Lehmer.
- Aliquot_sequence wikiPageWikiLink Divisor.
- Aliquot_sequence wikiPageWikiLink Divisor_function.
- Aliquot_sequence wikiPageWikiLink Eugène_Charles_Catalan.
- Aliquot_sequence wikiPageWikiLink Mathematics.
- Aliquot_sequence wikiPageWikiLink Perfect_number.
- Aliquot_sequence wikiPageWikiLink Periodic_function.
- Aliquot_sequence wikiPageWikiLink Prime_number.
- Aliquot_sequence wikiPageWikiLink Proper_divisor.
- Aliquot_sequence wikiPageWikiLink Recurrence_relation.
- Aliquot_sequence wikiPageWikiLink Recursive_sequence.
- Aliquot_sequence wikiPageWikiLink Sociable_number.
- Aliquot_sequence wikiPageWikiLinkText "Aliquot sequence".
- Aliquot_sequence wikiPageWikiLinkText "Catalan–Dickson conjecture on aliquot sequences".
- Aliquot_sequence wikiPageWikiLinkText "Lehmer five".
- Aliquot_sequence wikiPageWikiLinkText "aliquot sequence".
- Aliquot_sequence hasPhotoCollection Aliquot_sequence.
- Aliquot_sequence wikiPageUsesTemplate Template:As_of.
- Aliquot_sequence wikiPageUsesTemplate Template:Divisor_classes.
- Aliquot_sequence wikiPageUsesTemplate Template:OEIS.
- Aliquot_sequence wikiPageUsesTemplate Template:OEIS2C.
- Aliquot_sequence wikiPageUsesTemplate Template:Refbegin.
- Aliquot_sequence wikiPageUsesTemplate Template:Refend.
- Aliquot_sequence wikiPageUsesTemplate Template:Reflist.
- Aliquot_sequence subject Category:Arithmetic_functions.
- Aliquot_sequence subject Category:Divisor_function.
- Aliquot_sequence hypernym Sequence.
- Aliquot_sequence type Article.
- Aliquot_sequence type Type.
- Aliquot_sequence type Article.
- Aliquot_sequence type Function.
- Aliquot_sequence type Type.
- Aliquot_sequence comment "In mathematics, an aliquot sequence is a recursive sequence in which each term is the sum of the proper divisors of the previous term.".
- Aliquot_sequence label "Aliquot sequence".
- Aliquot_sequence sameAs متتالية_تجزيئية.
- Aliquot_sequence sameAs Alikvotfølge.
- Aliquot_sequence sameAs Inhaltskette.
- Aliquot_sequence sameAs Sucesión_alícuota.
- Aliquot_sequence sameAs Suite_aliquote.
- Aliquot_sequence sameAs סדרת_מחלקים.
- Aliquot_sequence sameAs アリコット数列.
- Aliquot_sequence sameAs m.02g8jf.
- Aliquot_sequence sameAs Аликвотная_последовательность.
- Aliquot_sequence sameAs Alikvotno_zaporedje.
- Aliquot_sequence sameAs Q1663510.
- Aliquot_sequence sameAs Q1663510.
- Aliquot_sequence sameAs 真因子和數列.
- Aliquot_sequence wasDerivedFrom Aliquot_sequence?oldid=666998490.
- Aliquot_sequence isPrimaryTopicOf Aliquot_sequence.