Sociable number

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

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

Number Theory