Sociable number
Encyclopedia
Sociable numbers are generalizations of the concepts of amicable number
s and perfect number
s. 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 number
s 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 following categorizes all known sociable numbers as of October 2009 by the length of the corresponding aliquot sequence:
Amicable number
Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number. A pair of amicable numbers constitutes an aliquot sequence of period 2...
s and perfect number
Perfect number
In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself . Equivalently, a perfect number is a number that is half the sum of all of its positive divisors i.e...
s. A set of sociable numbers is a kind of aliquot sequence
Aliquot sequence
In mathematics, an aliquot sequence is a recursive sequence in which each term is the sum of the proper divisors of the previous term. The aliquot sequence starting with a positive integer k can be defined formally in terms of the sum-of-divisors function σ1 in the following way:For example, the...
, or a sequence of numbers each of whose numbers is the sum of the factors
Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...
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
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
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 number
Amicable number
Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number. A pair of amicable numbers constitutes an aliquot sequence of period 2...
s 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
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
(and hence 1). Or equivalently, whether there exists a number whose aliquot sequence
Aliquot sequence
In mathematics, an aliquot sequence is a recursive sequence in which each term is the sum of the proper divisors of the previous term. The aliquot sequence starting with a positive integer k can be defined formally in terms of the sum-of-divisors function σ1 in the following way:For example, the...
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 |