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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 Lowest number in sequence
1 (perfect number) 43 6
2 (amicable number pair) 2X25E65E3 164
4 315X 50E8E8
5 1 7294
6 5 4214833623
8 4 266X43588
9 1 1X5E0X448
24 1 8350

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 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 (only list the numbers which belong to distinct families, e.g. 216 and 290 belong to the same families as 1E0, thus 216 and 290 are not listed here) (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. Also, the number X69 is notable as the least odd number such that the aliquot sequence does not repeat a term after 20 or fewer terms.

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

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