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, for 6 its proper factors are 1, 2, 3 - which add to 6 itself.
A pair of amicable numbers are then 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 are either sociable or end up at a prime (and hence 1), or whether conversely there exists a number whose aliquot sequence never terminates.
An example with period 4 (an example from the paper of Cohen) is
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