Sociable numbers are positive integers belonging to an aliquot sequence cycle of least period at least 3. 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 sequence 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, ....
Two early sociable cycles, of orders 5 and 28, were discovered by Poulet (1918). Moews (2026) credits nine groups of order 4 to Cohen (1970).
The smallest members of known sociable cycles include 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 sociable cycles as well as the number of such cycles known. The table records 152 known cycles of length greater than 2 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 with an integer 
and an initial positive
integer 
,
defined according to the generalized aliquot sequence
 |
(2)
|
The fixed points of this map satisfy 
,
so
 |
(3)
|
which gives a perfect number for 
and an 
-multiperfect number
for 
.
An initial value is called a 
-sociable number of order 
when every iterate is a positive
integer and 
is the least positive integer for which 
.
If 
and 
are distinct Mersenne primes, then
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
so 
and 
are 1/2-sociable numbers of order 2.
The following table lists the smallest members of some generalized 
-sociable cycles 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, 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.Cohen,
H. "On Amicable and Sociable Numbers." Math. Comput. 24,
423-429, 1970. 
." 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."