Aliquot sequence
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 can be defined formally in terms of the sum-of-divisors function or the aliquot sum function in the following way:If the condition is added, then the terms after 0 are all 0, and all aliquot sequences would be infinite, and we can conjecture that all aliquot sequences are either convergent or eventually periodic.
For example, the aliquot sequence of 10 is because:
Many aliquot sequences terminate at zero; all such sequences necessarily end with a prime number followed by 1, followed by 0. See for a list of such numbers up to 75. 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
- An amicable number has a repeating aliquot sequence of period 2. For instance, the aliquot sequence of 220 is
- A sociable number has a repeating aliquot sequence of period 3 or greater. For instance, the aliquot sequence of 1264460 is
- 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 Numbers like 95 that are not perfect, but have an eventually repeating aliquot sequence of period 1 are called aspiring numbers.
| Aliquot sequence of | Length | |
| 0 | 0 | 1 |
| 1 | 1, 0 | 2 |
| 2 | 2, 1, 0 | 3 |
| 3 | 3, 1, 0 | 3 |
| 4 | 4, 3, 1, 0 | 4 |
| 5 | 5, 1, 0 | 3 |
| 6 | 6 | 1 |
| 7 | 7, 1, 0 | 3 |
| 8 | 8, 7, 1, 0 | 4 |
| 9 | 9, 4, 3, 1, 0 | 5 |
| 10 | 10, 8, 7, 1, 0 | 5 |
| 11 | 11, 1, 0 | 3 |
| 12 | 12, 16, 15, 9, 4, 3, 1, 0 | 8 |
| 13 | 13, 1, 0 | 3 |
| 14 | 14, 10, 8, 7, 1, 0 | 6 |
| 15 | 15, 9, 4, 3, 1, 0 | 6 |
| 16 | 16, 15, 9, 4, 3, 1, 0 | 7 |
| 17 | 17, 1, 0 | 3 |
| 18 | 18, 21, 11, 1, 0 | 5 |
| 19 | 19, 1, 0 | 3 |
| 20 | 20, 22, 14, 10, 8, 7, 1, 0 | 8 |
| 21 | 21, 11, 1, 0 | 4 |
| 22 | 22, 14, 10, 8, 7, 1, 0 | 7 |
| 23 | 23, 1, 0 | 3 |
| 24 | 24, 36, 55, 17, 1, 0 | 6 |
| 25 | 25, 6 | 2 |
| 26 | 26, 16, 15, 9, 4, 3, 1, 0 | 8 |
| 27 | 27, 13, 1, 0 | 4 |
| 28 | 28 | 1 |
| 29 | 29, 1, 0 | 3 |
| 30 | 30, 42, 54, 66, 78, 90, 144, 259, 45, 33, 15, 9, 4, 3, 1, 0 | 16 |
| 31 | 31, 1, 0 | 3 |
| 32 | 32, 31, 1, 0 | 4 |
| 33 | 33, 15, 9, 4, 3, 1, 0 | 7 |
| 34 | 34, 20, 22, 14, 10, 8, 7, 1, 0 | 9 |
| 35 | 35, 13, 1, 0 | 4 |
| 36 | 36, 55, 17, 1, 0 | 5 |
| 37 | 37, 1, 0 | 3 |
| 38 | 38, 22, 14, 10, 8, 7, 1, 0 | 8 |
| 39 | 39, 17, 1, 0 | 4 |
| 40 | 40, 50, 43, 1, 0 | 5 |
| 41 | 41, 1, 0 | 3 |
| 42 | 42, 54, 66, 78, 90, 144, 259, 45, 33, 15, 9, 4, 3, 1, 0 | 15 |
| 43 | 43, 1, 0 | 3 |
| 44 | 44, 40, 50, 43, 1, 0 | 6 |
| 45 | 45, 33, 15, 9, 4, 3, 1, 0 | 8 |
| 46 | 46, 26, 16, 15, 9, 4, 3, 1, 0 | 9 |
| 47 | 47, 1, 0 | 3 |
The lengths of the aliquot sequences that start at are:
The final terms of the aliquot sequences that start at are:
Numbers whose aliquot sequence terminates in 1 are:
Numbers whose aliquot sequence does not terminate in 1 are:
Numbers whose aliquot sequence is known to terminate in a perfect number, other than perfect numbers themselves, are:
Numbers whose aliquot sequence terminates in a cycle with length at least 2 are:
Numbers whose aliquot sequence is not known to be finite or eventually periodic are:
A number that is never the successor in an aliquot sequence is called an untouchable number.
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 first five candidate numbers are often called the Lehmer five : 276, 552, 564, 660, and 966. While 276 may or may not reach a high apex in its aliquot sequence and then descend, the number 138 is notable for reaching a peak of 179,931,895,322 before returning to 1.In their lifetimes, Guy and Selfridge believed the Catalan–Dickson conjecture to be false: they conjecture some aliquot sequences are unbounded above.