Sociable number

In mathematics, sociable numbers are numbers whose aliquot sums form a cyclic sequence that begins and ends with the same number. They are generalizations of the concepts of amicable numbers and perfect numbers. The first two sociable sequences, or sociable chains, were discovered and named by the Belgian mathematician Paul Poulet in 1918. In a set of sociable numbers, each number is the sum of the proper factors of the preceding number, i.e., the sum excludes the preceding number itself. For the sequence to be sociable, the sequence must be cyclic and return 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, and searches for them have been made up to formula_1 as of 1970.
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 exist numbers whose aliquot sequence never terminates, and hence grows without bound.
Example.
An example with period 4:
List of known sociable numbers.
The following categorizes all known sociable numbers as of July 2018 by the length of the corresponding aliquot sequence:
It is conjectured that if "n" is congruent to 3 modulo 4 then there are no such sequence with length "n".
The 5-cycle sequence is: 12496, 14288, 15472, 14536, 14264
The only known 28-cycle is: 14316, 19116, 31704, 47616, 83328, 177792, 295488, 629072, 589786, 294896, 358336, 418904, 366556, 274924, 275444, 243760, 376736, 381028, 285778, 152990, 122410, 97946, 48976, 45946, 22976, 22744, 19916, 17716. . 
These two sequences provide the only sociable numbers below 1 million (other than the perfect and amicable numbers).
Searching for sociable numbers.
The aliquot sequence can be represented as a directed graph, formula_10, for a given integer formula_11, where formula_12 denotes the
sum of the proper divisors of formula_13.
Cycles in formula_10 represent sociable numbers within the interval formula_15. Two special cases are loops that represent perfect numbers and cycles of length two that represent amicable pairs.
Conjecture of the sum of sociable number cycles.
It is conjectured that as the number of sociable number cycles with length greater than 2 approaches infinity, the percentage of the sums of the sociable number cycles divisible by 10 approaches 100%. .