| Divisibility-based sets of integers |
| Forms of factorization: |
| Prime number |
| Composite number |
| Powerful number |
| Square-free number |
| Achilles number |
| Constrained divisor sums: |
| Perfect number |
| Almost perfect number |
| Quasiperfect number |
| Multiply perfect number |
| Hyperperfect number |
| Superperfect number |
| Unitary perfect number |
| Semiperfect number |
| Primitive semiperfect number |
| Practical number |
| Numbers with many divisors: |
| Abundant number |
| Highly abundant number |
| Superabundant number |
| Colossally abundant number |
| Highly composite number |
| Superior highly composite number |
| Other: |
| Untouchable number |
| Deficient number |
| Weird number |
| Amicable number |
| Friendly number |
| Sociable number |
| Solitary number |
| Sublime number |
| Harmonic divisor number |
| Frugal number |
| Equidigital number |
| Extravagant number |
| See also: |
| Divisor function |
| Divisor |
| Prime factor |
| Factorization |
Sociable numbers are generalizations of the concepts of amicable numbers and perfect numbers. A set of sociable numbers is a kind of aliquot sequence, or a sequence of numbers each of whose numbers is the sum of the factors of the preceding number, excluding the preceding number itself. For the sequence to be sociable, the sequence must be cyclic, eventually returning to its starting point.
The period of the sequence, or order of the set of sociable numbers, is the number of numbers in this cycle.
If the period of the sequence is 1, the number is a sociable number of order 1, or a perfect number—for example, the proper divisors of 6 are 1, 2, and 3, whose sum is again 6. A pair of amicable numbers is a set of sociable numbers of order 2. There are no known sociable numbers of order 3.
It is an open question whether all numbers end up at either a sociable number or at a prime (and hence 1). Or equivalently, whether there exists a number whose aliquot sequence never terminates.
An example with period 4:
- The sum of the proper divisors of 1264460 (22 * 5 * 17 * 3719) is:
- 1 + 2 + 4 + 5 + 10 + 17 + 20 + 34 + 68 + 85 + 170 + 340 + 3719 + 7438 + 14876 + 18595 + 37190 + 63223 + 74380 + 126446 + 252892 + 316115 + 632230 = 1547860
- The sum of the proper divisors of 1547860 (22 * 5 * 193 * 401) is:
- 1 + 2 + 4 + 5 + 10 + 20 + 193 + 386 + 401 + 772 + 802 + 965 + 1604 + 1930 + 2005 + 3860 + 4010 + 8020 + 77393 + 154786 + 309572 + 386965 + 773930 = 1727636
- The sum of the proper divisors of 1727636 (22 * 521 * 829) is:
- 1 + 2 + 4 + 521 + 829 + 1042 + 1658 + 2084 + 3316 + 431909 + 863818 = 1305184
- The sum of the proper divisors of 1305184 (25 * 40787) is:
- 1 + 2 + 4 + 8 + 16 + 32 + 40787 + 81574 + 163148 + 326296 + 652592 = 1264460.
The following categorizes all known sociable numbers as of October 2009 by the length of the corresponding aliquot sequence:
| Sequence
length |
Number of
sequences |
|---|---|
| 1
(perfect) |
47 |
| 2
(amicable) |
11,994,387 |
| 4 | 165 |
| 5 | 1 |
| 6 | 5 |
| 8 | 2 |
| 9 | 1 |
| 28 | 1 |
External links
- A list of known sociable numbers
- Extensive tables of perfect, amicable and sociable numbers
- Weisstein, Eric W., "Sociable numbers" from MathWorld.
References
- P. Poulet, #4865, L'intermediare des math. 25 (1918), pp. 100-101.
- H. Cohen, On amicable and sociable numbers, Math. Comp. 24 (1970), pp. 423-429
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