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Aliquot sequences
An aliquot sequence (or aliquot trajectory) is a recurrence in which each term is the sum of proper divisors of the previous term, where the initial term must be a positive integer, i.e.
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In the above,
s1(an ) := σ1(an ) − an |
σ1(an ) |
Contents
- eventually hits a prime number, which begets 1, and then the empty sum, i.e. 0 (and thus terminates, since the sum of proper divisors is defined for positive integers) (A080907);
- never hits a prime number (and thus never terminates), where it either
- cycles, with finite order
- 1: perfect numbers (e.g. {6, 6, ...}) (A000396);
- 2: amicable pairs (e.g. {220, 284, 220, ...}) (A063990);
- > 2: sociable numbers (e.g. {1264460, 1547860, 1727636, 1305184, 1264460, ...}) (A003416 Sociable numbers: smallest member of each cycle.);
- 1: perfect numbers (e.g.
- eventually cycles (by hitting a number belonging to a cycle), with finite order
- 1: aspiring numbers[1] (e.g. {25, 6, 6, ...}) (A063769);
- 2: (A??????);[2]
- > 2: (A??????);[3]
- 1: aspiring numbers[1] (e.g.
- never cycles, and is thus an infinite sequence of distinct composite terms, hence an unbounded sequence (this seems very unlikely, although the non-existence of such sequences has not yet been proved!).
- cycles, with finite order
An infinite unbounded aliquot sequence, growing monotonically or not, must manage to never hit any aliquot cycle whatsoever (no [even] or [odd?] perfect number, no member of an amicable pair, no member of social cycle) and it must never hit a prime number! This seems very unlikely: a term is obtained by “adding” the proper divisors of the previous term, so one would expect that hitting either a cycle or a prime number is a matter of probability (which is nonzero for each term), which would imply that a hit is eventually unavoidable (but there might be such sequences with any finite number of terms, although with decreasing probability as the number of terms increases... which corresponds to the [not yet established] truth of the Catalan–Dickson conjecture).
m < n |
m < n |
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Aliquot sequence of
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Length
A098007 | |||
---|---|---|---|---|---|
0 | 0 | 1 | |||
1 | 1, 0 | 2 = 1 + L (0) | |||
2 | 2, 1, 0 | 3 = 1 + L (1) | |||
3 | 3, 1, 0 | 3 = 1 + L (1) | |||
4 | 4, 3, 1, 0 | 4 = 1 + L (3) | |||
5 | 5, 1, 0 | 3 = 1 + L (1) | |||
6 | 6 | 1 | |||
7 | 7, 1, 0 | 3 = 1 + L (1) | |||
8 | 8, 7, 1, 0 | 4 = 1 + L (7) | |||
9 | 9, 4, 3, 1, 0 | 5 = 1 + L (4) | |||
10 | 10, 8, 7, 1, 0 | 5 = 1 + L (8) | |||
11 | 11, 1, 0 | 3 = 1 + L (1) | |||
12 | 12, 16, 15, 9, 4, 3, 1, 0 | 8 = 3 + L (9) | |||
13 | 13, 1, 0 | 3 = 1 + L (1) | |||
14 | 14, 10, 8, 7, 1, 0 | 6 = 1 + L (10) | |||
15 | 15, 9, 4, 3, 1, 0 | 6 = 1 + L (9) |
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Aliquot sequence of
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Length
A098007 | |||
---|---|---|---|---|---|
16 | 16, 15, 9, 4, 3, 1, 0 | 7 = 1 + L (15) | |||
17 | 17, 1, 0 | 3 = 1 + L (1) | |||
18 | 18, 21, 11, 1, 0 | 5 = 2 + L (11) | |||
19 | 19, 1, 0 | 3 = 1 + L (1) | |||
20 | 20, 22, 14, 10, 8, 7, 1, 0 | 8 = 2 + L (14) | |||
21 | 21, 11, 1, 0 | 4 = 1 + L (11) | |||
22 | 22, 14, 10, 8, 7, 1, 0 | 7 = 1 + L (14) | |||
23 | 23, 1, 0 | 3 = 1 + L (1) | |||
24 | 24, 36, 55, 17, 1, 0 | 6 = 3 + L (17) | |||
25 | 25, 6 | 2 = 1 + L (6) | |||
26 | 26, 16, 15, 9, 4, 3, 1, 0 | 8 = 1 + L (16) | |||
27 | 27, 13, 1, 0 | 4 = 1 + L (13) | |||
28 | 28 | 1 | |||
29 | 29, 1, 0 | 3 = 1 + L (1) | |||
30 | 30, 42, 54, 66, 78, 90, 144, 259, 45, 33, 15, 9, 4, 3, 1, 0 | 16 = 10 + L (15) | |||
31 | 31, 1, 0 | 3 = 1 + L (1) |
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Aliquot sequence of
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Length
A098007 | |||
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32 | 32, 31, 1, 0 | 4 = 1 + L (31) | |||
33 | 33, 15, 9, 4, 3, 1, 0 | 7 = 1 + L (15) | |||
34 | 34, 20, 22, 14, 10, 8, 7, 1, 0 | 9 = 1 + L (20) | |||
35 | 35, 13, 1, 0 | 4 = 1 + L (13) | |||
36 | 36, 55, 17, 1, 0 | 5 = 2 + L (17) | |||
37 | 37, 1, 0 | 3 = 1 + L (1) | |||
38 | 38, 22, 14, 10, 8, 7, 1, 0 | 8 = 1 + L (22) | |||
39 | 39, 17, 1, 0 | 4 = 1 + L (17) | |||
40 | 40, 50, 43, 1, 0 | 5 = 3 + L (1) | |||
41 | 41, 1, 0 | 3 = 1 + L (1) | |||
42 | 42, 54, 66, 78, 90, 144, 259, 45, 33, 15, 9, 4, 3, 1, 0 | 15 = 8 + L (33) | |||
43 | 43, 1, 0 | 3 = 1 + L (1) | |||
44 | 44, 40, 50, 43, 1, 0 | 6 = 1 + L (40) | |||
45 | 45, 33, 15, 9, 4, 3, 1, 0 | 8 = 1 + L (33) | |||
46 | 46, 26, 16, 15, 9, 4, 3, 1, 0 | 9 = 1 + L (26) | |||
47 | 47, 1, 0 | 3 = 1 + L (1) |
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Aliquot sequence of
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Length
A098007 | |||
---|---|---|---|---|---|
48 | 48, 76, 64, 63, 41, 1, 0 | 7 = 4 + L (41) | |||
49 | 49, 8, 7, 1, 0 | 5 = 1 + L (8) | |||
50 | 50, 43, 1, 0 | 4 = 1 + L (43) | |||
51 | 51, 21, 11, 1, 0 | 5 = 1 + L (21) | |||
52 | 52, 46, 26, 16, 15, 9, 4, 3, 1, 0 | 10 = 1 + L (46) | |||
53 | 53, 1, 0 | 3 = 1 + L (1) | |||
54 | 54, 66, 78, 90, 144, 259, 45, 33, 15, 9, 4, 3, 1, 0 | 14 = 6 + L (45) | |||
55 | 55, 17, 1, 0 | 4 = 1 + L (17) | |||
56 | 56, 64, 63, 41, 1, 0 | 6 = 3 + L (41) | |||
57 | 57, 23, 1, 0 | 4 = 1 + L (23) | |||
58 | 58, 32, 31, 1, 0 | 5 = 1 + L (32) | |||
59 | 59, 1, 0 | 3 = 1 + L (1) | |||
60 | 60, 108, 172, 136, 134, 70, 74, 40, 50, 43, 1, 0 | 12 = 7 + L (40) | |||
61 | 61, 1, 0 | 3 = 1 + L (1) | |||
62 | 62, 34, 20, 22, 14, 10, 8, 7, 1, 0 | 10 = 1 + L (34) | |||
63 | 63, 41, 1, 0 | 4 = 1 + L (41) |
Catalan’s aliquot sequence conjecture
In 1888, Eugène Charles Catalan proposed the [not yet established and probably false] conjecture[4]
Conjecture (Catalan’s aliquot sequence conjecture, 1888). (Eugene Catalan)
Every sequence contains “either unity or a perfect number.”
The conjecture was amended by Leonard Eugene Dickson[4]
Conjecture (Catalan–Dickson conjecture). (Leonard Eugene Dickson)
No unbounded aliquot sequences exist (“every non-periodic chain contains a prime”).*
* Which means that every aliquot sequence either terminates in a prime number followed by 1, is an aliquot cycle (a perfect number, a friendly pair or an aliquot cycle of sociable numbers) or eventually reaches an aliquot cycle.
Paul Erdős and Hendrik Lenstra postulated that unbounded aliquot sequences do exist.[4]
Lehmer five
The Lehmer five (named after Derrick Henry Lehmer), 276, 552, 564, 660 and 966 may potentially have an unbounded aliquot sequence (while they may also lead to aliquot sequence terms so large that their prime factorization becomes intractable, while a ridiculously large prime number or some member of an aliquot cycle (e.g. a ridiculously large perfect number, amicable number or sociable number), lies on the trajectory, after a ridiculously large number of iterations...).[5]
Sequences
A098007 Length of aliquot sequence forn |
- {2, 3, 3, 4, 3, 1, 3, 4, 5, 5, 3, 8, 3, 6, 6, 7, 3, 5, 3, 8, 4, 7, 3, 6, 2, 8, 4, 1, 3, 16, 3, 4, 7, 9, 4, 5, 3, 8, 4, 5, 3, 15, 3, 6, 8, 9, 3, 7, 5, 4, 5, 10, 3, 14, 4, 6, 4, 5, 3, 12, 3, 10, 4, 5, 4, 13, 3, 6, 5, 7, 3, 10, 3, 6, ...}
n |
- {1, 2, 2, 3, 2, 0, 2, 3, 4, 4, 2, 7, 2, 5, 5, 6, 2, 4, 2, 7, 3, 6, 2, 5, 1, 7, 3, 0, 2, 15, 2, 3, 6, 8, 3, 4, 2, 7, 3, 4, 2, 14, 2, 5, 7, 8, 2, 6, 4, 3, 4, 9, 2, 13, 3, 5, 3, 4, 2, 11, 2, 9, 3, 4, 3, 12, 2, 5, 4, 6, 2, 9, 2, 5, 5, 5, ...}
A080907 Numbers whose aliquot sequence terminates in a 1. ({1} ∪ {primes} ∪ {A127162})
- {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, 51, ...}
A127162 Composite numbers whose aliquot sequences terminate by encountering a prime number.
- {4, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, ...}
A126016 Numbers whose aliquot sequence does not terminate in 1. (Necessarily composite numbers.)
- {6, 25, 28, 95, 119, 143, 220, ...}&{276?, 284, 306?, 396?, 417, 445, 496, ...}
n |
n > 1 |
- {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, 11, 3, ...}
n |
n |
n |
- {6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, 2658455991569831744654692615953842176, ...}
A063769 Aspiring numbers: numbers whose aliquot sequence terminates in a perfect number.
- {25, 95, 119, 143, 417, 445, 565, 608, 650, 652, 675, 685, 783, 790, 909, 913, ...}
A063990 Amicable numbers.
- {220, 284, 1184, 1210, 2620, 2924, 5020, 5564, 6232, 6368, 10744, 10856, 12285, 14595, 17296, 18416, 63020, 66928, 66992, 67095, 69615, 71145, 76084, 79750, 87633, 88730, 100485, ...}
A?????? Numbers whose aliquot sequence terminates in an aliquot cycle of amicable numbers.[2]
- {?, ...}
A003416 Sociable numbers: smallest member of each cycle.
- {12496, 14316, 1264460, 2115324, 2784580, 4938136, 7169104, 18048976, 18656380, 28158165, 46722700, 81128632, 174277820, 209524210, 330003580, 498215416, 805984760, 1095447416, ...}
A?????? Numbers whose aliquot sequence terminates in an aliquot cycle of sociable numbers.[3]
- {?, ...}
A121507 Conjectured list of numbers whose aliquot sequence eventually reaches a cycle of length two or more.
- {220, 284, 562, 1064, 1184, 1188, 1210, 1308, 1336, 1380, 1420, 1490, 1604, 1690, 1692, 1772, 1816, 1898, 2008, 2122, 2152, 2172, 2362, 2542, 2620, 2630, 2652, 2676, 2678, 2856, 2924, 2930, ...}
A121508 Conjectured list of numbers whose aliquot sequence eventually reaches a cycle of length two or more, but which are not themselves part of the cycle.
- {562, 1064, 1188, 1308, 1336, 1380, 1420, 1490, 1604, 1690, 1692, 1772, 1816, 1898, 2008, 2122, 2152, 2172, 2362, 2542, 2630, 2652, 2676, 2678, 2856, 2930, 2950, 2974, 3124, 3162, 3202, 3278, ...}
n |
- {2, 5, 52, 88, 96, 120, 124, 146, 162, 188, 206, 210, 216, 238, 246, 248, 262, 268, 276, 288, 290, 292, 304, 306, 322, 324, 326, 336, 342, 372, 406, 408, 426, 430, 448, 472, 474, 498, 516, 518, 520, 530, ...}
See also
Notes
- ↑ Weisstein, Eric W., Aspiring Number, from MathWorld—A Wolfram Web Resource.
- ↑ 2.0 2.1 “Aspiring amicable numbers”? Have any been found?
- ↑ 3.0 3.1 “Aspiring sociable numbers”? Have any been found?
- ↑ 4.0 4.1 4.2 Joy Heysse, Aliquot Sequences, 2010.
- ↑ While one may succeed in proving that a given aliquot sequence never reaches a prime or an [even] perfect number, there would still remain the possibility of reaching an amicable number or a sociable number. (Assuming also that no odd perfect numbers exist!)
External links
- Guy, Richard K.; Selfridge, J. L. (1975). “What Drives an Aliquot Sequence?”. Mathematics of Computation 29 (129): pp. 101–107 .