Sociable numbers are numbers that result in a periodic aliquot sequence, where an aliquot sequence is the sequence of numbers obtained by repeatedly applying the restricted divisor function
 |
(1)
|
to 
and 
is the usual divisor function.
If the period of the aliquot cycle is 1, the number is called a perfect number. If the period is 2, the two numbers are called an amicable
pair. In general, if the period is 
, the number is called sociable of order 
. For example, 1264460 is a sociable number of order four since
its aliquot sequence is 1264460, 1547860, 1727636,
1305184, 1264460, ....
Only two groups of sociable numbers were known prior to 1970, namely the sets of orders 5 and 28 discovered by Poulet (1918). In 1970, Cohen discovered nine groups of order 4.
The first few sociable numbers are 12496, 14316, 1264460, 2115324, 2784580, 4938136, ... (OEIS A003416), which have orders 5, 28, 4, 4, 4, 4, ... (OEIS A052470). The following table summarizes the smallest members of known social cycles as well as the number of such cycles known (Moews). Excluding perfect numbers, a total of 152 sociable cycles are known as of Feb. 2009 (Pedersen).
 | OEIS | # |  -sociable
numbers |
| 4 | A090615 | 142 | 1264460, 2115324, 2784580, 4938136, 7169104, 18048976, 18656380, ... |
| 5 | 1 | 12496 | |
| 6 | A119478 | 5 | 21548919483, 90632826380, 1771417411016, 3524434872392, 4773123705616 |
| 8 | 2 | 1095447416, 1276254780 | |
| 9 | 1 | 805984760 | |
| 28 | 1 | 14316 |
Y. Kohmoto has considered a generalization of the sociable numbers defined according to the generalized aliquot sequence
 |
(2)
|
Multiperfect numbers are fixed points of this mapping, since if 
,
then
 |
(3)
|
which is the definition of an 
-multiperfect number. If the sequence 
becomes cyclic after 
terms, it is then called an 
-sociable number of order 
.
If 
and 
are distinct Mersenne primes, then
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
so 
and 
are 1/2-sociable numbers of order 2.
The following table summarizes the smallest members of the generalized 
-aliquot sequences of order 
, found by Kohmoto.
 |  | starting numbers |
| 3 | 2 | 14913024 |
| 4 | 2 | 2096640, 422688000 |
| 4 | 12 | 3396556800 |
-fach iterierten Teilerersummenfunktion." Mitt. Math.
Gesellsch. Hamburg 9, 34-48, 1969.Cohen, H. "On Amicable
and Sociable Numbers." Math. Comput. 24, 423-429, 1970.Cohen,
G. L. and te Riele, H. J. J. "Iterating the Sum-of-Divisors Function."
Amsterdam, Netherlands: Centrum voor Wiskunde en Informatica Report NM-R9525 1995.Creyaufmüller,
W. "Aliquot Sequences." 
." Math. Comput. 57, 849-855, 1991.Moews,
D. and Moews, P. C. "A Search for Aliquot Cycles and Amicable Pairs."
Math. Comput. 61, 935-938, 1993.Pedersen, J. A. M.
"Tables of Aliquot Cycles."