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{{short description|Mathematical recursive sequence}} {{short description|Mathematical recursive sequence}}
{{unsolved|mathematics|Do all aliquot sequences eventually end with a prime number, a perfect number, or a set of amicable or sociable numbers? (Catalan's aliquot sequence conjecture)}} {{unsolved|mathematics|Do all aliquot sequences eventually end with a prime number, a perfect number, or a set of amicable or sociable numbers? (Catalan's aliquot sequence conjecture)}}

In ], an '''aliquot sequence''' is a sequence of positive integers in which each term is the sum of the ]s of the previous term. If the sequence reaches the number 1, it ends, since the sum of the proper divisors of 1 is 0. In ], an '''aliquot sequence''' is a sequence of positive integers in which each term is the sum of the ]s of the previous term. If the sequence reaches the number 1, it ends, since the sum of the proper divisors of 1 is 0.


==Definition and overview== ==Definition and overview==
The aliquot sequence starting with a positive integer ''k'' can be defined formally in terms of the ] σ<sub>1</sub> or the ] function ''s'' in the following way:<ref name="mw">{{MathWorld | urlname=AliquotSequence | title=Aliquot Sequence}}</ref> The aliquot sequence starting with a positive integer ''k'' can be defined formally in terms of the ] σ<sub>1</sub> or the ] function ''s'' in the following way:

: ''s''<sub>0</sub> = ''k'' : ''s''<sub>0</sub> = ''k''
: ''s''<sub>n</sub> = ''s''(''s''<sub>''n''&minus;1</sub>) = σ<sub>1</sub>(''s''<sub>''n''&minus;1</sub>) &minus; ''s''<sub>''n''&minus;1</sub> if ''s''<sub>''n''&minus;1</sub> &gt; 0 : ''s''<sub>n</sub> = ''s''(''s''<sub>''n''&minus;1</sub>) = σ<sub>1</sub>(''s''<sub>''n''&minus;1</sub>) &minus; ''s''<sub>''n''&minus;1</sub> if ''s''<sub>''n''&minus;1</sub> &gt; 0
: ''s''<sub>n</sub> = 0 if ''s''<sub>''n''&minus;1</sub> = 0 ---> (if we add this condition, then the terms after 0 are all 0, and all Aliquot sequences would be infinite sequence, and we can conjecture that all Aliquot sequences are ], the limit of these sequences are usually 0 or 6) : ''s''<sub>n</sub> = 0 if ''s''<sub>''n''&minus;1</sub> = 0 ---> (if we add this condition, then the terms after 0 are all 0, and all Aliquot sequences would be infinite sequence, and we can conjecture that all Aliquot sequences are ], the limit of these sequences are usually 0 or 6)
and ''s''(0) is undefined.


and ''s''(0) is undefined or should be infinity, or 1+2+3+4+5+6+7+8+9+X+E+10+...=−0.1 (since all integers are divisors of 0, and all nonzero integers are proper divisors of 0).
For example, the aliquot sequence of 10 is 10, 8, 7, 1, 0 because:

For example, the aliquot sequence of 10 is 10, 14, 13, 9, 4, 3, 1, 0 because:


:σ<sub>1</sub>(10) &minus; 10 = 5 + 2 + 1 = 8, :σ<sub>1</sub>(10) &minus; 10 = 6 + 4 + 3 + 2 + 1 = 14,
:σ<sub>1</sub>(8) &minus; 8 = 4 + 2 + 1 = 7, :σ<sub>1</sub>(14) &minus; 14 = 8 + 4 + 2 + 1 = 13,
:σ<sub>1</sub>(7) &minus; 7 = 1, :σ<sub>1</sub>(13) &minus; 13 = 5 + 3 + 1 = 9,
:σ<sub>1</sub>(9) &minus; 9 = 3 + 1 = 4,
:σ<sub>1</sub>(4) &minus; 4 = 2 + 1 = 3,
:σ<sub>1</sub>(3) &minus; 3 = 1,
:σ<sub>1</sub>(1) &minus; 1 = 0. :σ<sub>1</sub>(1) &minus; 1 = 0.


Many aliquot sequences terminate at zero; all such sequences necessarily end with a ] followed by 1 (since the only proper divisor of a prime is 1), followed by 0 (since 1 has no proper divisors). See {{OEIS|id=A080907}} for a list of such numbers up to 75. There are a variety of ways in which an aliquot sequence might not terminate: Many aliquot sequences terminate at zero; all such sequences necessarily end with a ] 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 ] has a repeating aliquot sequence of period 1. The aliquot sequence of 6, for example, is 6, 6, 6, 6, ...
* An ] has a repeating aliquot sequence of period 2. For instance, the aliquot sequence of 220 is 220, 284, 220, 284, ... * A ] has a repeating aliquot sequence of period 1. e.g., the aliquot sequence of 6 is 6, 6, 6, 6, 6, 6, 6, 6,6, 6, 6, 6, ...

* A ] 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, ...
* An ] has a repeating aliquot sequence of period 2. e.g., the aliquot sequence of 164 is 164, 1E8, 164, 1E8, 164, 1E8, 164, 1E8, 164, 1E8, 164, 1E8, ...
* 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'''.<ref>{{Cite OEIS|1=A063769|2=Aspiring numbers: numbers whose aliquot sequence terminates in a perfect number. }}</ref>

* A ] has a repeating aliquot sequence of period 3 or greater (sometimes the term ''sociable number'' is used to encompass amicable numbers as well). e.g., the aliquot sequence of 50E8E8 is 50E8E8, 627904, 6E3958, 52E394, 50E8E8, 627904, 6E3958, 52E394, 50E8E8, 627904, 6E3958, 52E394, ...

The perfect numbers are

:6, 24, 354, 4854, E29E854, 17E8891054, 22777E33854, 1057E377XX9481E854, 65933E8691303X4485432649E134E87854, 2638X61964528067X1E505XXE7082227479878068X38891054, 2978485903026X5E727X6426597X88X711339953E2149326851X90891054, 41949486714E83298784387X55153E8X52E812EE45E2E899EX8X3624683384306733854, ...

The amicable number pairs are

: (164, 1E8), (828, 84X), (1624, 1838), (2XX4, 3278), (3734, 3828), (6274, 6348), (7139, 8543), (X014, X7X8), (30578, 38044), (32894, 32928), (329E3, 35209), (34353, 42869), (3X19X, 43422), (4X199, 5EX23), (5X909, 68XE3), (5X994, 5E328), (69E94, 74788), (6X432, 8178X), (83554, 86068), (86014, 887X8), (8E334, 99888), (95X18, 991X4), (116424, 157338), (12X724, 169798), (134E12, 1890XX), (152308, 1732E4), (1911X8, 19E614), (1X7518, 1E542X), (203154, 209X68), (212399, 214423), (24E548, 283674), (254E74, 291048), (261274, 294048), (267X08, 2X41E4), (270374, 275848), (282678, 319544), (2E0248, 325074), (328659, 357XX3), (365114, 3757X8), (373974, 3E3848), (398623, 463599), (401734, 423788), (43E942, 45027X), (478102, 4941EX), (47959X, 524022), (488169, 5948E3), (491E94, 520788), (519099, 547723), (54095X, 5E6862), (556283, 5E8539), (571928, 5X7694), (5X8572, 70364X), (5X9884, 704338), (60XE62, 65105X), (686472, 83574X), (729023, 762399), (845168, 849X54), (8EX38X, 991832), (X7E188, E9X434), (XE4328, E67294), (XE715X, 1044X62), (E01588, E92634), (E31928, EX1694), (E32538, 12E2684), ...

The sociable number cycles (length>2) are

:(7294, 8328, 8E54, 84E4, 8308), (8350, E090, 16420, 23680, 40280, 86X80, 123000, 264068, 24538X, 1227X8, 153454, 182508, 158164, 113124, 113498, E9094, 162028, 164604, 11946X, 74652, 5XX0X, 48822, 24414, 2270X, 11368, 111E4, E638, X304), (50E8E8, 627904, 6E3958, 52E394), (860190, 113EEX4, 1280018, E2XEX4), (E23544, 1116018, 12X9704, 11666E8), (17X1874, 1E15948, 1X15274, 1996648), (2498954, 2636798, 293XE34, 2628128), (6065014, 68941X8, 6402214, 65453X8), (62E8624, 6X58238, 9708384, 8174838), (951E299, X020923, X3EE299, X054923), (13792724, 17049698, 15EX6E24, 17957798), (23205534, 266E8388, 28344834, 2673X388), (4X447278, 58X89944, 73X50278, 5X79E944), (5X20452X, 6X737692, 65612E2X, 650X9092), (92626364, X169E8E8, E1886E44, 103434478), ...


{|class="wikitable" {|class="wikitable"
! Length of sociable number cycle
|''n''
! Number of known such cycles
|Aliquot sequence of ''n''
|length ({{oeis|A098007}})
|''n''
|Aliquot sequence of ''n''
|length ({{oeis|A098007}})
|''n''
|Aliquot sequence of ''n''
|length ({{oeis|A098007}})
|''n''
|Aliquot sequence of ''n''
|length ({{oeis|A098007}})
|- |-
|1 (perfect number)
|0
|0 |43
|1
|12
|12, 16, 15, 9, 4, 3, 1, 0
|8
|24
|24, 36, 55, 17, 1, 0
|6
|36
|36, 55, 17, 1, 0
|5
|- |-
|2 (amicable number pair)
|1
|2X25E65E3
|1, 0
|2
|13
|13, 1, 0
|3
|25
|25, 6
|2
|37
|37, 1, 0
|3
|- |-
|2
|2, 1, 0
|3
|14
|14, 10, 8, 7, 1, 0
|6
|26
|26, 16, 15, 9, 4, 3, 1, 0
|8
|38
|38, 22, 14, 10, 8, 7, 1, 0
|8
|-
|3
|3, 1, 0
|3
|15
|15, 9, 4, 3, 1, 0
|6
|27
|27, 13, 1, 0
|4
|39
|39, 17, 1, 0
|4 |4
|315X
|- |-
|4 |5
|4, 3, 1, 0
|4
|16
|16, 15, 9, 4, 3, 1, 0
|7
|28
|28
|1 |1
|40
|40, 50, 43, 1, 0
|5
|- |-
|5
|5, 1, 0
|3
|17
|17, 1, 0
|3
|29
|29, 1, 0
|3
|41
|41, 1, 0
|3
|-
|6
|6 |6
|1
|18
|18, 21, 11, 1, 0
|5 |5
|30
|30, 42, 54, 66, 78, 90, 144, 259, 45, 33, 15, 9, 4, 3, 1, 0
|16
|42
|42, 54, 66, 78, 90, 144, 259, 45, 33, 15, 9, 4, 3, 1, 0
|15
|-
|7
|7, 1, 0
|3
|19
|19, 1, 0
|3
|31
|31, 1, 0
|3
|43
|43, 1, 0
|3
|- |-
|8 |8
|8, 7, 1, 0
|4 |4
|20
|20, 22, 14, 10, 8, 7, 1, 0
|8
|32
|32, 31, 1, 0
|4
|44
|44, 40, 50, 43, 1, 0
|6
|- |-
|9 |9
|1
|9, 4, 3, 1, 0
|5
|21
|21, 11, 1, 0
|4
|33
|33, 15, 9, 4, 3, 1, 0
|7
|45
|45, 33, 15, 9, 4, 3, 1, 0
|8
|- |-
|10 |24
|1
|10, 8, 7, 1, 0
|5 |}

|22
It is conjectured that there are no sociable number cycles with length 3
|22, 14, 10, 8, 7, 1, 0

|7
It is conjectured that the numbers in all sociable number cycles (including amicable number pairs) are either all even or all odd.
|34

|34, 20, 22, 14, 10, 8, 7, 1, 0
The number 24 is not only the length of the longest known sociable number cycle, but also the length of the largest known ].
|9

|46
== Table of Aliquot sequences with start values up to 100 ==
|46, 26, 16, 15, 9, 4, 3, 1, 0

|9
{|class="wikitable"
||''n''||Aliquot sequence of ''n''||''n''||Aliquot sequence of ''n''||''n''||Aliquot sequence of ''n''||''n''||Aliquot sequence of ''n''
|- |-
||1||1, 0, ||31||31, 1, 0, ||61||61, 1, 0, ||91||91, 1, 0,
|11
|-
|11, 1, 0
||2||2, 1, 0, ||32||32, 1X, 12, X, 8, 7, 1, 0, ||62||62, 34, 42, 37, 1, 0, ||92||92, 8X, 48, 54, 53, 35, 1, 0,
|3
|23 |-
||3||3, 1, 0, ||33||33, 15, 1, 0, ||63||63, 41, 8, 7, 1, 0, ||93||93, 35, 1, 0,
|23, 1, 0
|3 |-
||4||4, 3, 1, 0, ||34||34, 42, 37, 1, 0, ||64||64, 54, 53, 35, 1, 0, ||94||94, E4, E2, 5X, 62, 34, 42, 37, 1, 0,
|35
|-
|35, 13, 1, 0
||5||5, 1, 0, ||35||35, 1, 0, ||65||65, 17, 1, 0, ||95||95, 1, 0,
|4
|47 |-
||6||6, ||36||36, 46, 56, 66, 76, 100, 197, 39, 29, 13, 9, 4, 3, 1, 0, ||66||66, 76, 100, 197, 39, 29, 13, 9, 4, 3, 1, 0, ||96||96, X6, 136, 146, 1X6, 316, 533, 289, E3, 89, 73, 29, 13, 9, 4, 3, 1, 0,
|47, 1, 0
|3 |-
||7||7, 1, 0, ||37||37, 1, 0, ||67||67, 1, 0, ||97||97, 25, 1, 0,
|-
||8||8, 7, 1, 0, ||38||38, 34, 42, 37, 1, 0, ||68||68, 8X, 48, 54, 53, 35, 1, 0, ||98||98, 7X, 42, 37, 1, 0,
|-
||9||9, 4, 3, 1, 0, ||39||39, 29, 13, 9, 4, 3, 1, 0, ||69||69, 34, 42, 37, 1, 0, ||99||99, 55, 17, 1, 0,
|-
||X||X, 8, 7, 1, 0, ||3X||3X, 22, 14, 13, 9, 4, 3, 1, 0, ||6X||6X, 38, 34, 42, 37, 1, 0, ||9X||9X, 52, 2X, 18, 1X, 12, X, 8, 7, 1, 0,
|-
||E||E, 1, 0, ||3E||3E, 1, 0, ||6E||6E, 1, 0, ||9E||9E, 21, 6,
|-
||10||10, 14, 13, 9, 4, 3, 1, 0, ||40||40, 64, 54, 53, 35, 1, 0, ||70||70, E8, 144, 14E, 31, 1, 0, ||X0||X0, 180, 360, 740, 1180, 1X60, 4516, 8200, 16X23, 13999, 7055, 1, 0,
|-
||11||11, 1, 0, ||41||41, 8, 7, 1, 0, ||71||71, 1E, 1, 0, ||X1||X1, 10, 14, 13, 9, 4, 3, 1, 0,
|-
||12||12, X, 8, 7, 1, 0, ||42||42, 37, 1, 0, ||72||72, 3X, 22, 14, 13, 9, 4, 3, 1, 0, ||X2||X2, 54, 53, 35, 1, 0,
|-
||13||13, 9, 4, 3, 1, 0, ||43||43, 19, E, 1, 0, ||73||73, 29, 13, 9, 4, 3, 1, 0, ||X3||X3, 39, 29, 13, 9, 4, 3, 1, 0,
|-
||14||14, 13, 9, 4, 3, 1, 0, ||44||44, 3X, 22, 14, 13, 9, 4, 3, 1, 0, ||74||74, 78, 64, 54, 53, 35, 1, 0, ||X4||X4, 84, 99, 55, 17, 1, 0,
|-
||15||15, 1, 0, ||45||45, 1, 0, ||75||75, 1, 0, ||X5||X5, 27, 1, 0,
|-
||16||16, 19, E, 1, 0, ||46||46, 56, 66, 76, 100, 197, 39, 29, 13, 9, 4, 3, 1, 0, ||76||76, 100, 197, 39, 29, 13, 9, 4, 3, 1, 0, ||X6||X6, 136, 146, 1X6, 316, 533, 289, E3, 89, 73, 29, 13, 9, 4, 3, 1, 0,
|-
||17||17, 1, 0, ||47||47, 15, 1, 0, ||77||77, 19, E, 1, 0, ||X7||X7, 1, 0,
|-
||18||18, 1X, 12, X, 8, 7, 1, 0, ||48||48, 54, 53, 35, 1, 0, ||78||78, 64, 54, 53, 35, 1, 0, ||X8||X8, X7, 1, 0,
|-
||19||19, E, 1, 0, ||49||49, 1E, 1, 0, ||79||79, 2E, 11, 1, 0, ||X9||X9, 3E, 1, 0,
|-
||1X||1X, 12, X, 8, 7, 1, 0, ||4X||4X, 28, 27, 1, 0, ||7X||7X, 42, 37, 1, 0, ||XX||XX, X2, 54, 53, 35, 1, 0,
|-
||1E||1E, 1, 0, ||4E||4E, 1, 0, ||7E||7E, 21, 6, ||XE||XE, 1, 0,
|-
||20||20, 30, 47, 15, 1, 0, ||50||50, 90, 124, E4, E2, 5X, 62, 34, 42, 37, 1, 0, ||80||80, 110, 178, 134, 128, 144, 14E, 31, 1, 0, ||E0||E0, 150, 210, 3E4, 368, 367, 69, 34, 42, 37, 1, 0,
|-
||21||21, 6, ||51||51, 1, 0, ||81||81, 1, 0, ||E1||E1, 23, 11, 1, 0,
|-
||22||22, 14, 13, 9, 4, 3, 1, 0, ||52||52, 2X, 18, 1X, 12, X, 8, 7, 1, 0, ||82||82, 61, 1, 0, ||E2||E2, 5X, 62, 34, 42, 37, 1, 0,
|-
||23||23, 11, 1, 0, ||53||53, 35, 1, 0, ||83||83, 49, 1E, 1, 0, ||E3||E3, 89, 73, 29, 13, 9, 4, 3, 1, 0,
|-
||24||24, ||54||54, 53, 35, 1, 0, ||84||84, 99, 55, 17, 1, 0, ||E4||E4, E2, 5X, 62, 34, 42, 37, 1, 0,
|-
||25||25, 1, 0, ||55||55, 17, 1, 0, ||85||85, 1, 0, ||E5||E5, 1, 0,
|-
||26||26, 36, 46, 56, 66, 76, 100, 197, 39, 29, 13, 9, 4, 3, 1, 0, ||56||56, 66, 76, 100, 197, 39, 29, 13, 9, 4, 3, 1, 0, ||86||86, 96, X6, 136, 146, 1X6, 316, 533, 289, E3, 89, 73, 29, 13, 9, 4, 3, 1, 0, ||E6||E6, 106, 166, 176, 220, 380, 680, 1260, 2216, 32X6, 3976, 6276, 7806, 8006, E2E6, 10106, 10606, 12636, 13586, 13596, 1E4X6, 30716, 38966, 54116, 832X6, 102816, 123190, 272X30, 43E750, 5X9370, 7X44E8, 858E44, 771552, 3X004X, 283172, 15074X, 86388, 103094, 12X880, 2026X0, 303X80, 4E9840, 989954, XX1368, 12E8E94, 1283258, E11864, 853858, 935374, 9X5048, 10XE2E4, 130X908, 1123504, XE39X4, 1024918, 92X6E4, 1001508, 19356E4, 2955908, 39128E4, 5188708, 651X954, 9806508, 8565764, 65EX4E0, 8X86110, 116E1310, 1619E4E0, 20253290, 415E48E0, 645E0E90, 98X47270, 122209EX4, E3555918, 858651E4, 81357908, 886112E4, 82946108, 813E4054, 7E98429X, 42997022, 214X9614, 3029E7X8, 29E7X244, 216508X8, 2159E19X, 12241722, 7315X9X, 432EE82, 253003X, 1286622, 75564X, 388928, 38E754, 40X6X0, 626820, 98E960, 1852220, 29XX080, 47115X0, 6X78320, E0E48X0, 17178720, 293974X0, 6X464720, 1172834X0, 189849E20, 272672E20, 3X99XX520, 5XX1196X0, 96E853920, 1681582020, 28950902X0, 412269X920, 79XE7752X0, E8X5551X80, 1747XE89100, 2XX569E22X2, 1642617977X, 9E65517E42, 55E509244X, 35E7704972, 3455XE484X, 1828E58428, 1771E14904, 128454069X, 1074068122, 7595E3X9X, 40651XE22, 23982109X, 15909E822, 8X6X199X, 55684E22, 2X16244X, 1619E372, 1481744X, 8423372, 602830X, 3889552, 2E9520X, 178E222, 106939X, 63E422, 5E079X, 2E63E2, 22120X, 1309E2, 10080X, 617E2, 3454X, 18288, 18104, 1309X, 8122, 4074, 6948, 8644, E378, 9724, X998, 814X, 4088, 368X, 2162, 145X, 832, 98X, 4X8, 584, 668, 644, 49X, 252, 28X, 148, 1X1, 4E, 1, 0,
|-
||27||27, 1, 0, ||57||57, 1, 0, ||87||87, 1, 0, ||E7||E7, 1, 0,
|-
||28||28, 27, 1, 0, ||58||58, 4X, 28, 27, 1, 0, ||88||88, 8X, 48, 54, 53, 35, 1, 0, ||E8||E8, 144, 14E, 31, 1, 0,
|-
||29||29, 13, 9, 4, 3, 1, 0, ||59||59, 23, 11, 1, 0, ||89||89, 73, 29, 13, 9, 4, 3, 1, 0, ||E9||E9, 43, 19, E, 1, 0,
|-
||2X||2X, 18, 1X, 12, X, 8, 7, 1, 0, ||5X||5X, 62, 34, 42, 37, 1, 0, ||8X||8X, 48, 54, 53, 35, 1, 0, ||EX||EX, 62, 34, 42, 37, 1, 0,
|-
||2E||2E, 11, 1, 0, ||5E||5E, 1, 0, ||8E||8E, 1, 0, ||EE||EE, 21, 6,
|-
||30||30, 47, 15, 1, 0, ||60||60, X3, 39, 29, 13, 9, 4, 3, 1, 0, ||90||90, 124, E4, E2, 5X, 62, 34, 42, 37, 1, 0, ||100||100, 197, 39, 29, 13, 9, 4, 3, 1, 0,
|} |}


The lengths of the Aliquot sequences that start at ''n'' are The lengths of the Aliquot sequences that start at ''n'' are
:1, 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, ... {{OEIS|id=A044050}}


:1, 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, 13, 2, 3, 6, 8, 3, 4, 2, 7, 3, 4, 2, 12, 2, 5, 7, 8, 2, 6, 4, 3, 4, 9, 2, 11, 3, 5, 3, 4, 2, E, 2, 9, 3, 4, 3, 10, 2, 5, 4, 6, 2, 9, 2, 5, 5, 5, 3, E, 2, 7, 5, 6, 2, 6, 3, 9, 7, 7, 2, X, 4, 6, 4, 4, 2, 9, 2, 3, 4, 5, 2, 16, 2, 7, 8, 6, 2, X, 2, 7, 3, 9, 2, 15, 3, 5, 4, X, 2, 10, 8, 5, 8, 6, 3, 14, 2, 3, 3, 6, 2, E, 4, 7, 9, 8, 2, 12X, 2, 5, 5, 6, 2, 9, ...
The final terms (excluding 1) of the Aliquot sequences that start at ''n'' are

:1, 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, ... {{OEIS|id=A115350}}
The maximum terms of the Aliquot sequences that start at ''n'' are

:1, 2, 3, 4, 5, 6, 7, 8, 9, X, E, 14, 11, 12, 13, 14, 15, 19, 17, 1X, 19, 1X, 1E, 47, 21, 22, 23, 24, 25, 197, 27, 28, 29, 2X, 2E, 47, 31, 32, 33, 42, 35, 197, 37, 42, 39, 3X, 3E, 64, 41, 42, 43, 44, 45, 197, 47, 54, 49, 4X, 4E, 124, 51, 52, 53, 54, 55, 197, 57, 58, 59, 62, 5E, X3, 61, 62, 63, 64, 65, 197, 67, 8X, 69, 6X, 6E, 14E, 71, 72, 73, 78, 75, 197, 77, 78, 79, 7X, 7E, 178, 81, 82, 83, 99, 85, 533, 87, 8X, 89, 8X, 8E, 124, 91, 92, 93, E4, 95, 533, 97, 98, 99, 9X, 9E, 16X23, X1, X2, X3, X4, X5, 533, X7, X8, X9, XX, XE, 3E4, E1, E2, E3, E4, E5, 2XX569E22X2, E7, 14E, E9, EX, EE, 197, ...

The final terms (excluding 1 and 0) of the Aliquot sequences that start at ''n'' are

:1, 2, 3, 3, 5, 6, 7, 7, 3, 7, E, 3, 11, 7, 3, 3, 15, E, 17, 7, E, 7, 1E, 15, 6, 3, 11, 24, 25, 3, 27, 27, 3, 7, 11, 15, 31, 7, 15, 37, 35, 3, 37, 37, 3, 3, 3E, 35, 7, 37, E, 3, 45, 3, 15, 35, 1E, 27, 4E, 37, 51, 7, 35, 35, 17, 3, 57, 27, 11, 37, 5E, 3, 61, 37, 7, 35, 17, 3, 67, 35, 37, 37, 6E, 31, 1E, 3, 3, 35, 75, 3, E, 35, 11, 37, 6, 31, 81, 61, 1E, 17, 85, 3, 87, 35, 3, 35, 8E, 37, 91, 35, 35, 37, 95, 3, 25, 37, 17, 7, 6, 7055, 3, 35, 3, 17, 27, 3, X7, X7, 3E, 35, XE, 37, 11, 37, 3, 37, E5, 4E, E7, 31, E, 37, 6, 3, ...


Numbers whose Aliquot sequence terminates in 1 are Numbers whose Aliquot sequence terminates in 1 are
:1, 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, ... {{OEIS|id=A080907}}


:1, 2, 3, 4, 5, 7, 8, 9, X, E, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1X, 1E, 20, 22, 23, 25, 26, 27, 28, 29, 2X, 2E, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 3X, 3E, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 4X, 4E, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 5X, 5E, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 6X, 6E, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 7X, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 8X, 8E, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 9X, X0, X1, X2, X3, X4, X5, X6, X7, X8, X9, XX, XE, E0, E1, E2, E3, E4, E5, E6, E7, E8, E9, EX, 100, ...
Numbers whose Aliquot sequence known to terminate in a ], other than perfect numbers themselves (6, 28, 496, ...), are

:25, 95, 119, 143, 417, 445, 565, 608, 650, 652, 675, 685, 783, 790, 909, 913, ... {{OEIS|id=A063769}}
Numbers whose Aliquot sequence terminates in a perfect number are

:6, 21, 24, 7E, 9E, EE, 2X9, 311, 354, 3E1, 428, 462, 464, 483, 491, 553, 55X, 639, 641, 821, 86E, 8EX, X01, X03, XE2, E0E, E41, EXE, EE1, ...


Numbers whose Aliquot sequence terminates in a cycle with length at least 2 are Numbers whose Aliquot sequence terminates in a cycle with length at least 2 are

:220, 284, 562, 1064, 1184, 1188, 1210, 1308, 1336, 1380, 1420, 1490, 1604, 1690, 1692, 1772, 1816, 1898, 2008, 2122, 2152, 2172, 2362, ... {{OEIS|id=A121507}}
:164, 1E8, 3XX, 748, 828, 830, 84X, 910, 934, 970, 9X4, X42, E18, E8X, E90, ...


Numbers whose Aliquot sequence is not known to be finite or eventually periodic are Numbers whose Aliquot sequence is not known to be finite or eventually periodic are
:276, 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, ... {{OEIS|id=A131884}}


:1E0, 216, 290, 3X0, 3E0, 470, 4X0, 550, 590, 620, 686, 6E0, 756, 766, 776, 780, 7X6, 856, 906, 916, 920, 936, 946, 950, 980, 986, 996, X20, X30, X40, X60, XX0, XE0, XE6, E06, E40, E56, E66, E76, EE6, ...
A number that is never the successor in an aliquot sequence is called an ].
:], ], ], ], ], ], ], ], ], ], ], ], ], ], ], ], 262, 268, ], ], ], 292, 304, 306, 322, 324, 326, 336, 342, 372, 406, 408, 426, 430, 448, 472, 474, 498, ... {{OEIS|id=A005114}}


A number that is never the successor in an aliquot sequence is called an untouchable number
== Catalan-Dickson conjecture ==
An important ] due to ], sometimes called the Catalan–] conjecture, 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.<ref>{{MathWorld | urlname=CatalansAliquotSequenceConjecture | title=Catalan's Aliquot Sequence Conjecture}}</ref> The alternative would be that a number exists whose aliquot sequence is infinite yet never repeats. 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 often called the '''Lehmer five''' (named after ]): ], 552, 564, 660, and 966.<ref>{{cite web|url=http://www.aliquot.de/lehmer.htm|title=Lehmer Five|first=Wolfgang|last=Creyaufmüller|date=May 24, 2014|access-date=June 14, 2015}}</ref> However, it is worth noting that 276 may reach a high apex in its aliquot sequence and then descend; the number 138 reaches a peak of 179931895322 before returning to 1.


:2, 5, 44, 74, 80, X0, X4, 102, 116, 138, 152, 156, 160, 17X, 186, 188, 19X, 1X4, 1E0, 200, 202, 204, 214, 216, 22X, 230, 232, 240, 246, 270, 29X, 2X0, 2E6, 2EX, 314, 334, 336, 356, 370, 372, 374, 382, 390, 3X0, 3X4, 3XX, 400, 408, 430, 440, 442, 444, 46X, 478, 47X, 4E0, 4E6, 4EX, 506, 510, 516, 524, 526, 530, 53X, 540, 552, 554, 560, 56X, 570, 582, 598, 5X8, 5E0, 608, 624, 626, 628, 62X, 632, 652, 65X, 660, 684, 686, 694, 69X, 6E0, 6E6, 718, 730, 732, 744, 750, 756, 75X, 760, 77X, 790, 7X0, 7X6, 7E6, 7E8, 7EX, 808, 80X, 814, 824, 82X, 834, 840, 850, 85X, 870, 87X, 880, 886, 888, 88X, 896, 8X0, 8E4, 900, 914, 916, 918, 91X, 926, 930, 93X, 942, 944, 954, 970, 978, 986, 990, 992, 9X2, 9X4, 9X6, 9EX, X30, X56, X58, X5X, X6X, X74, X82, X86, XX6, XE6, E04, E10, E40, E4X, E56, E80, E82, E90, EE0, EE2, 1000, ...
] and ] believe the Catalan–Dickson conjecture is false (so they conjecture some aliquot sequences are ] above (or diverge)).<ref>A. S. Mosunov, </ref>


== Catalan-Dickson conjecture ==
{{As of|2015|04|alt=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.<ref>{{cite web|url=http://www.aliquot.de/aliquote.htm|title=Aliquot Pages|first=Wolfgang|last=Creyaufmüller|date=April 29, 2015|access-date=June 14, 2015}}</ref>
An important ] due to ], sometimes called the Catalan–] conjecture, 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 never repeats. Any one of the many numbers whose aliquot sequences have not been fully determined might be such a number. The candidate numbers up to 1000 are 1E0, 3X0, 3E0, 470, 686, 756, 7X6, X20, X30, X40, X60, XX0, XE6, E40 (most of these numbers end with 0). However, it is worth noting that 1E0 may reach a high apex in its aliquot sequence and then descend; the number E6 reaches a peak of 2XX569E22X2 before returning to 1.

== Systematically searching for aliquot sequences ==


] and ] believe the Catalan–Dickson conjecture is false (so they conjecture some aliquot sequences are ] above (or diverge)).
The aliquot sequence can be represented as a ], <math>G_{n,s}</math>, for a given integer <math>n</math>, where <math>s(k)</math> denotes the
sum of the proper divisors of <math>k</math>.<ref>{{citation|title=Distributed cycle detection in large-scale sparse graphs|first1=Rodrigo Caetano|last1=Rocha|first2=Bhalchandra|last2=Thatte|year=2015|publisher=Simpósio Brasileiro de Pesquisa Operacional (SBPO)|doi=10.13140/RG.2.1.1233.8640}}</ref>
] in <math>G_{n,s}</math> represent sociable numbers within the interval <math></math>. Two special cases are loops that represent ] and cycles of length two that represent ].


== See also == == See also ==

Revision as of 04:00, 24 August 2020

Mathematical recursive sequence Unsolved problem in mathematics: Do all aliquot sequences eventually end with a prime number, a perfect number, or a set of amicable or sociable numbers? (Catalan's aliquot sequence conjecture) (more unsolved problems in mathematics)

In mathematics, an aliquot sequence is a sequence of positive integers in which each term is the sum of the proper divisors of the previous term. If the sequence reaches the number 1, it ends, since the sum of the proper divisors of 1 is 0.

Definition and overview

The aliquot sequence starting with a positive integer k can be defined formally in terms of the sum-of-divisors function σ1 or the aliquot sum function s in the following way:

s0 = k
sn = s(sn−1) = σ1(sn−1) − sn−1 if sn−1 > 0
sn = 0 if sn−1 = 0 ---> (if we add this condition, then the terms after 0 are all 0, and all Aliquot sequences would be infinite sequence, and we can conjecture that all Aliquot sequences are convergent, the limit of these sequences are usually 0 or 6)

and s(0) is undefined or should be infinity, or 1+2+3+4+5+6+7+8+9+X+E+10+...=−0.1 (since all integers are divisors of 0, and all nonzero integers are proper divisors of 0).

For example, the aliquot sequence of 10 is 10, 14, 13, 9, 4, 3, 1, 0 because:

σ1(10) − 10 = 6 + 4 + 3 + 2 + 1 = 14,
σ1(14) − 14 = 8 + 4 + 2 + 1 = 13,
σ1(13) − 13 = 5 + 3 + 1 = 9,
σ1(9) − 9 = 3 + 1 = 4,
σ1(4) − 4 = 2 + 1 = 3,
σ1(3) − 3 = 1,
σ1(1) − 1 = 0.

Many aliquot sequences terminate at zero; 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. e.g., the aliquot sequence of 6 is 6, 6, 6, 6, 6, 6, 6, 6,6, 6, 6, 6, ...
  • An amicable number has a repeating aliquot sequence of period 2. e.g., the aliquot sequence of 164 is 164, 1E8, 164, 1E8, 164, 1E8, 164, 1E8, 164, 1E8, 164, 1E8, ...
  • 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). e.g., the aliquot sequence of 50E8E8 is 50E8E8, 627904, 6E3958, 52E394, 50E8E8, 627904, 6E3958, 52E394, 50E8E8, 627904, 6E3958, 52E394, ...

The perfect numbers are

6, 24, 354, 4854, E29E854, 17E8891054, 22777E33854, 1057E377XX9481E854, 65933E8691303X4485432649E134E87854, 2638X61964528067X1E505XXE7082227479878068X38891054, 2978485903026X5E727X6426597X88X711339953E2149326851X90891054, 41949486714E83298784387X55153E8X52E812EE45E2E899EX8X3624683384306733854, ...

The amicable number pairs are

(164, 1E8), (828, 84X), (1624, 1838), (2XX4, 3278), (3734, 3828), (6274, 6348), (7139, 8543), (X014, X7X8), (30578, 38044), (32894, 32928), (329E3, 35209), (34353, 42869), (3X19X, 43422), (4X199, 5EX23), (5X909, 68XE3), (5X994, 5E328), (69E94, 74788), (6X432, 8178X), (83554, 86068), (86014, 887X8), (8E334, 99888), (95X18, 991X4), (116424, 157338), (12X724, 169798), (134E12, 1890XX), (152308, 1732E4), (1911X8, 19E614), (1X7518, 1E542X), (203154, 209X68), (212399, 214423), (24E548, 283674), (254E74, 291048), (261274, 294048), (267X08, 2X41E4), (270374, 275848), (282678, 319544), (2E0248, 325074), (328659, 357XX3), (365114, 3757X8), (373974, 3E3848), (398623, 463599), (401734, 423788), (43E942, 45027X), (478102, 4941EX), (47959X, 524022), (488169, 5948E3), (491E94, 520788), (519099, 547723), (54095X, 5E6862), (556283, 5E8539), (571928, 5X7694), (5X8572, 70364X), (5X9884, 704338), (60XE62, 65105X), (686472, 83574X), (729023, 762399), (845168, 849X54), (8EX38X, 991832), (X7E188, E9X434), (XE4328, E67294), (XE715X, 1044X62), (E01588, E92634), (E31928, EX1694), (E32538, 12E2684), ...

The sociable number cycles (length>2) are

(7294, 8328, 8E54, 84E4, 8308), (8350, E090, 16420, 23680, 40280, 86X80, 123000, 264068, 24538X, 1227X8, 153454, 182508, 158164, 113124, 113498, E9094, 162028, 164604, 11946X, 74652, 5XX0X, 48822, 24414, 2270X, 11368, 111E4, E638, X304), (50E8E8, 627904, 6E3958, 52E394), (860190, 113EEX4, 1280018, E2XEX4), (E23544, 1116018, 12X9704, 11666E8), (17X1874, 1E15948, 1X15274, 1996648), (2498954, 2636798, 293XE34, 2628128), (6065014, 68941X8, 6402214, 65453X8), (62E8624, 6X58238, 9708384, 8174838), (951E299, X020923, X3EE299, X054923), (13792724, 17049698, 15EX6E24, 17957798), (23205534, 266E8388, 28344834, 2673X388), (4X447278, 58X89944, 73X50278, 5X79E944), (5X20452X, 6X737692, 65612E2X, 650X9092), (92626364, X169E8E8, E1886E44, 103434478), ...
Length of sociable number cycle Number of known such cycles
1 (perfect number) 43
2 (amicable number pair) 2X25E65E3
4 315X
5 1
6 5
8 4
9 1
24 1

It is conjectured that there are no sociable number cycles with length 3

It is conjectured that the numbers in all sociable number cycles (including amicable number pairs) are either all even or all odd.

The number 24 is not only the length of the longest known sociable number cycle, but also the length of the largest known polydivisible number.

Table of Aliquot sequences with start values up to 100

n Aliquot sequence of n n Aliquot sequence of n n Aliquot sequence of n n Aliquot sequence of n
1 1, 0, 31 31, 1, 0, 61 61, 1, 0, 91 91, 1, 0,
2 2, 1, 0, 32 32, 1X, 12, X, 8, 7, 1, 0, 62 62, 34, 42, 37, 1, 0, 92 92, 8X, 48, 54, 53, 35, 1, 0,
3 3, 1, 0, 33 33, 15, 1, 0, 63 63, 41, 8, 7, 1, 0, 93 93, 35, 1, 0,
4 4, 3, 1, 0, 34 34, 42, 37, 1, 0, 64 64, 54, 53, 35, 1, 0, 94 94, E4, E2, 5X, 62, 34, 42, 37, 1, 0,
5 5, 1, 0, 35 35, 1, 0, 65 65, 17, 1, 0, 95 95, 1, 0,
6 6, 36 36, 46, 56, 66, 76, 100, 197, 39, 29, 13, 9, 4, 3, 1, 0, 66 66, 76, 100, 197, 39, 29, 13, 9, 4, 3, 1, 0, 96 96, X6, 136, 146, 1X6, 316, 533, 289, E3, 89, 73, 29, 13, 9, 4, 3, 1, 0,
7 7, 1, 0, 37 37, 1, 0, 67 67, 1, 0, 97 97, 25, 1, 0,
8 8, 7, 1, 0, 38 38, 34, 42, 37, 1, 0, 68 68, 8X, 48, 54, 53, 35, 1, 0, 98 98, 7X, 42, 37, 1, 0,
9 9, 4, 3, 1, 0, 39 39, 29, 13, 9, 4, 3, 1, 0, 69 69, 34, 42, 37, 1, 0, 99 99, 55, 17, 1, 0,
X X, 8, 7, 1, 0, 3X 3X, 22, 14, 13, 9, 4, 3, 1, 0, 6X 6X, 38, 34, 42, 37, 1, 0, 9X 9X, 52, 2X, 18, 1X, 12, X, 8, 7, 1, 0,
E E, 1, 0, 3E 3E, 1, 0, 6E 6E, 1, 0, 9E 9E, 21, 6,
10 10, 14, 13, 9, 4, 3, 1, 0, 40 40, 64, 54, 53, 35, 1, 0, 70 70, E8, 144, 14E, 31, 1, 0, X0 X0, 180, 360, 740, 1180, 1X60, 4516, 8200, 16X23, 13999, 7055, 1, 0,
11 11, 1, 0, 41 41, 8, 7, 1, 0, 71 71, 1E, 1, 0, X1 X1, 10, 14, 13, 9, 4, 3, 1, 0,
12 12, X, 8, 7, 1, 0, 42 42, 37, 1, 0, 72 72, 3X, 22, 14, 13, 9, 4, 3, 1, 0, X2 X2, 54, 53, 35, 1, 0,
13 13, 9, 4, 3, 1, 0, 43 43, 19, E, 1, 0, 73 73, 29, 13, 9, 4, 3, 1, 0, X3 X3, 39, 29, 13, 9, 4, 3, 1, 0,
14 14, 13, 9, 4, 3, 1, 0, 44 44, 3X, 22, 14, 13, 9, 4, 3, 1, 0, 74 74, 78, 64, 54, 53, 35, 1, 0, X4 X4, 84, 99, 55, 17, 1, 0,
15 15, 1, 0, 45 45, 1, 0, 75 75, 1, 0, X5 X5, 27, 1, 0,
16 16, 19, E, 1, 0, 46 46, 56, 66, 76, 100, 197, 39, 29, 13, 9, 4, 3, 1, 0, 76 76, 100, 197, 39, 29, 13, 9, 4, 3, 1, 0, X6 X6, 136, 146, 1X6, 316, 533, 289, E3, 89, 73, 29, 13, 9, 4, 3, 1, 0,
17 17, 1, 0, 47 47, 15, 1, 0, 77 77, 19, E, 1, 0, X7 X7, 1, 0,
18 18, 1X, 12, X, 8, 7, 1, 0, 48 48, 54, 53, 35, 1, 0, 78 78, 64, 54, 53, 35, 1, 0, X8 X8, X7, 1, 0,
19 19, E, 1, 0, 49 49, 1E, 1, 0, 79 79, 2E, 11, 1, 0, X9 X9, 3E, 1, 0,
1X 1X, 12, X, 8, 7, 1, 0, 4X 4X, 28, 27, 1, 0, 7X 7X, 42, 37, 1, 0, XX XX, X2, 54, 53, 35, 1, 0,
1E 1E, 1, 0, 4E 4E, 1, 0, 7E 7E, 21, 6, XE XE, 1, 0,
20 20, 30, 47, 15, 1, 0, 50 50, 90, 124, E4, E2, 5X, 62, 34, 42, 37, 1, 0, 80 80, 110, 178, 134, 128, 144, 14E, 31, 1, 0, E0 E0, 150, 210, 3E4, 368, 367, 69, 34, 42, 37, 1, 0,
21 21, 6, 51 51, 1, 0, 81 81, 1, 0, E1 E1, 23, 11, 1, 0,
22 22, 14, 13, 9, 4, 3, 1, 0, 52 52, 2X, 18, 1X, 12, X, 8, 7, 1, 0, 82 82, 61, 1, 0, E2 E2, 5X, 62, 34, 42, 37, 1, 0,
23 23, 11, 1, 0, 53 53, 35, 1, 0, 83 83, 49, 1E, 1, 0, E3 E3, 89, 73, 29, 13, 9, 4, 3, 1, 0,
24 24, 54 54, 53, 35, 1, 0, 84 84, 99, 55, 17, 1, 0, E4 E4, E2, 5X, 62, 34, 42, 37, 1, 0,
25 25, 1, 0, 55 55, 17, 1, 0, 85 85, 1, 0, E5 E5, 1, 0,
26 26, 36, 46, 56, 66, 76, 100, 197, 39, 29, 13, 9, 4, 3, 1, 0, 56 56, 66, 76, 100, 197, 39, 29, 13, 9, 4, 3, 1, 0, 86 86, 96, X6, 136, 146, 1X6, 316, 533, 289, E3, 89, 73, 29, 13, 9, 4, 3, 1, 0, E6 E6, 106, 166, 176, 220, 380, 680, 1260, 2216, 32X6, 3976, 6276, 7806, 8006, E2E6, 10106, 10606, 12636, 13586, 13596, 1E4X6, 30716, 38966, 54116, 832X6, 102816, 123190, 272X30, 43E750, 5X9370, 7X44E8, 858E44, 771552, 3X004X, 283172, 15074X, 86388, 103094, 12X880, 2026X0, 303X80, 4E9840, 989954, XX1368, 12E8E94, 1283258, E11864, 853858, 935374, 9X5048, 10XE2E4, 130X908, 1123504, XE39X4, 1024918, 92X6E4, 1001508, 19356E4, 2955908, 39128E4, 5188708, 651X954, 9806508, 8565764, 65EX4E0, 8X86110, 116E1310, 1619E4E0, 20253290, 415E48E0, 645E0E90, 98X47270, 122209EX4, E3555918, 858651E4, 81357908, 886112E4, 82946108, 813E4054, 7E98429X, 42997022, 214X9614, 3029E7X8, 29E7X244, 216508X8, 2159E19X, 12241722, 7315X9X, 432EE82, 253003X, 1286622, 75564X, 388928, 38E754, 40X6X0, 626820, 98E960, 1852220, 29XX080, 47115X0, 6X78320, E0E48X0, 17178720, 293974X0, 6X464720, 1172834X0, 189849E20, 272672E20, 3X99XX520, 5XX1196X0, 96E853920, 1681582020, 28950902X0, 412269X920, 79XE7752X0, E8X5551X80, 1747XE89100, 2XX569E22X2, 1642617977X, 9E65517E42, 55E509244X, 35E7704972, 3455XE484X, 1828E58428, 1771E14904, 128454069X, 1074068122, 7595E3X9X, 40651XE22, 23982109X, 15909E822, 8X6X199X, 55684E22, 2X16244X, 1619E372, 1481744X, 8423372, 602830X, 3889552, 2E9520X, 178E222, 106939X, 63E422, 5E079X, 2E63E2, 22120X, 1309E2, 10080X, 617E2, 3454X, 18288, 18104, 1309X, 8122, 4074, 6948, 8644, E378, 9724, X998, 814X, 4088, 368X, 2162, 145X, 832, 98X, 4X8, 584, 668, 644, 49X, 252, 28X, 148, 1X1, 4E, 1, 0,
27 27, 1, 0, 57 57, 1, 0, 87 87, 1, 0, E7 E7, 1, 0,
28 28, 27, 1, 0, 58 58, 4X, 28, 27, 1, 0, 88 88, 8X, 48, 54, 53, 35, 1, 0, E8 E8, 144, 14E, 31, 1, 0,
29 29, 13, 9, 4, 3, 1, 0, 59 59, 23, 11, 1, 0, 89 89, 73, 29, 13, 9, 4, 3, 1, 0, E9 E9, 43, 19, E, 1, 0,
2X 2X, 18, 1X, 12, X, 8, 7, 1, 0, 5X 5X, 62, 34, 42, 37, 1, 0, 8X 8X, 48, 54, 53, 35, 1, 0, EX EX, 62, 34, 42, 37, 1, 0,
2E 2E, 11, 1, 0, 5E 5E, 1, 0, 8E 8E, 1, 0, EE EE, 21, 6,
30 30, 47, 15, 1, 0, 60 60, X3, 39, 29, 13, 9, 4, 3, 1, 0, 90 90, 124, E4, E2, 5X, 62, 34, 42, 37, 1, 0, 100 100, 197, 39, 29, 13, 9, 4, 3, 1, 0,

The lengths of the Aliquot sequences that start at n are

1, 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, 13, 2, 3, 6, 8, 3, 4, 2, 7, 3, 4, 2, 12, 2, 5, 7, 8, 2, 6, 4, 3, 4, 9, 2, 11, 3, 5, 3, 4, 2, E, 2, 9, 3, 4, 3, 10, 2, 5, 4, 6, 2, 9, 2, 5, 5, 5, 3, E, 2, 7, 5, 6, 2, 6, 3, 9, 7, 7, 2, X, 4, 6, 4, 4, 2, 9, 2, 3, 4, 5, 2, 16, 2, 7, 8, 6, 2, X, 2, 7, 3, 9, 2, 15, 3, 5, 4, X, 2, 10, 8, 5, 8, 6, 3, 14, 2, 3, 3, 6, 2, E, 4, 7, 9, 8, 2, 12X, 2, 5, 5, 6, 2, 9, ...

The maximum terms of the Aliquot sequences that start at n are

1, 2, 3, 4, 5, 6, 7, 8, 9, X, E, 14, 11, 12, 13, 14, 15, 19, 17, 1X, 19, 1X, 1E, 47, 21, 22, 23, 24, 25, 197, 27, 28, 29, 2X, 2E, 47, 31, 32, 33, 42, 35, 197, 37, 42, 39, 3X, 3E, 64, 41, 42, 43, 44, 45, 197, 47, 54, 49, 4X, 4E, 124, 51, 52, 53, 54, 55, 197, 57, 58, 59, 62, 5E, X3, 61, 62, 63, 64, 65, 197, 67, 8X, 69, 6X, 6E, 14E, 71, 72, 73, 78, 75, 197, 77, 78, 79, 7X, 7E, 178, 81, 82, 83, 99, 85, 533, 87, 8X, 89, 8X, 8E, 124, 91, 92, 93, E4, 95, 533, 97, 98, 99, 9X, 9E, 16X23, X1, X2, X3, X4, X5, 533, X7, X8, X9, XX, XE, 3E4, E1, E2, E3, E4, E5, 2XX569E22X2, E7, 14E, E9, EX, EE, 197, ...

The final terms (excluding 1 and 0) of the Aliquot sequences that start at n are

1, 2, 3, 3, 5, 6, 7, 7, 3, 7, E, 3, 11, 7, 3, 3, 15, E, 17, 7, E, 7, 1E, 15, 6, 3, 11, 24, 25, 3, 27, 27, 3, 7, 11, 15, 31, 7, 15, 37, 35, 3, 37, 37, 3, 3, 3E, 35, 7, 37, E, 3, 45, 3, 15, 35, 1E, 27, 4E, 37, 51, 7, 35, 35, 17, 3, 57, 27, 11, 37, 5E, 3, 61, 37, 7, 35, 17, 3, 67, 35, 37, 37, 6E, 31, 1E, 3, 3, 35, 75, 3, E, 35, 11, 37, 6, 31, 81, 61, 1E, 17, 85, 3, 87, 35, 3, 35, 8E, 37, 91, 35, 35, 37, 95, 3, 25, 37, 17, 7, 6, 7055, 3, 35, 3, 17, 27, 3, X7, X7, 3E, 35, XE, 37, 11, 37, 3, 37, E5, 4E, E7, 31, E, 37, 6, 3, ...

Numbers whose Aliquot sequence terminates in 1 are

1, 2, 3, 4, 5, 7, 8, 9, X, E, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1X, 1E, 20, 22, 23, 25, 26, 27, 28, 29, 2X, 2E, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 3X, 3E, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 4X, 4E, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 5X, 5E, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 6X, 6E, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 7X, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 8X, 8E, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 9X, X0, X1, X2, X3, X4, X5, X6, X7, X8, X9, XX, XE, E0, E1, E2, E3, E4, E5, E6, E7, E8, E9, EX, 100, ...

Numbers whose Aliquot sequence terminates in a perfect number are

6, 21, 24, 7E, 9E, EE, 2X9, 311, 354, 3E1, 428, 462, 464, 483, 491, 553, 55X, 639, 641, 821, 86E, 8EX, X01, X03, XE2, E0E, E41, EXE, EE1, ...

Numbers whose Aliquot sequence terminates in a cycle with length at least 2 are

164, 1E8, 3XX, 748, 828, 830, 84X, 910, 934, 970, 9X4, X42, E18, E8X, E90, ...

Numbers whose Aliquot sequence is not known to be finite or eventually periodic are

1E0, 216, 290, 3X0, 3E0, 470, 4X0, 550, 590, 620, 686, 6E0, 756, 766, 776, 780, 7X6, 856, 906, 916, 920, 936, 946, 950, 980, 986, 996, X20, X30, X40, X60, XX0, XE0, XE6, E06, E40, E56, E66, E76, EE6, ...

A number that is never the successor in an aliquot sequence is called an untouchable number

2, 5, 44, 74, 80, X0, X4, 102, 116, 138, 152, 156, 160, 17X, 186, 188, 19X, 1X4, 1E0, 200, 202, 204, 214, 216, 22X, 230, 232, 240, 246, 270, 29X, 2X0, 2E6, 2EX, 314, 334, 336, 356, 370, 372, 374, 382, 390, 3X0, 3X4, 3XX, 400, 408, 430, 440, 442, 444, 46X, 478, 47X, 4E0, 4E6, 4EX, 506, 510, 516, 524, 526, 530, 53X, 540, 552, 554, 560, 56X, 570, 582, 598, 5X8, 5E0, 608, 624, 626, 628, 62X, 632, 652, 65X, 660, 684, 686, 694, 69X, 6E0, 6E6, 718, 730, 732, 744, 750, 756, 75X, 760, 77X, 790, 7X0, 7X6, 7E6, 7E8, 7EX, 808, 80X, 814, 824, 82X, 834, 840, 850, 85X, 870, 87X, 880, 886, 888, 88X, 896, 8X0, 8E4, 900, 914, 916, 918, 91X, 926, 930, 93X, 942, 944, 954, 970, 978, 986, 990, 992, 9X2, 9X4, 9X6, 9EX, X30, X56, X58, X5X, X6X, X74, X82, X86, XX6, XE6, E04, E10, E40, E4X, E56, E80, E82, E90, EE0, EE2, 1000, ...

Catalan-Dickson conjecture

An important conjecture due to Catalan, sometimes called the Catalan–Dickson conjecture, 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 never repeats. Any one of the many numbers whose aliquot sequences have not been fully determined might be such a number. The candidate numbers up to 1000 are 1E0, 3X0, 3E0, 470, 686, 756, 7X6, X20, X30, X40, X60, XX0, XE6, E40 (most of these numbers end with 0). However, it is worth noting that 1E0 may reach a high apex in its aliquot sequence and then descend; the number E6 reaches a peak of 2XX569E22X2 before returning to 1.

Guy and Selfridge believe the Catalan–Dickson conjecture is false (so they conjecture some aliquot sequences are unbounded above (or diverge)).

See also

Notes

References

  • Manuel Benito; Wolfgang Creyaufmüller; Juan Luis Varona; Paul Zimmermann. Aliquot Sequence 3630 Ends After Reaching 100 Digits. Experimental Mathematics, vol. 11, num. 2, Natick, MA, 2002, p. 201-206.
  • W. Creyaufmüller. Primzahlfamilien - Das Catalan'sche Problem und die Familien der Primzahlen im Bereich 1 bis 3000 im Detail. Stuttgart 2000 (3rd ed.), 327p.

External links

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Overview Divisibility of 60
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