Editing Sociable number (section)

{{short description|Numbers whose aliquot sums form a cyclic sequence}}
In [[mathematics]], '''sociable numbers''' are numbers whose [[Aliquot sum#Definition|aliquot sums]] form a [[periodic sequence]]. They are generalizations of the concepts of [[perfect number]]s and [[amicable number]]s. The first two sociable sequences, or sociable chains, were discovered and named by the [[Belgium|Belgian]] [[mathematician]] [[Paul Poulet (mathematician)|Paul Poulet]] in 1918.<ref>P. Poulet, #4865, [[L'Intermédiaire des Mathématiciens]] '''25''' (1918), pp.&nbsp;100–101. (The full text can be found at [https://proofwiki.org/wiki/Catalan-Dickson_Conjecture ProofWiki: Catalan-Dickson Conjecture].)</ref> In a sociable sequence, each number is the sum of the [[proper divisors]] 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 [[Periodic function|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 divisor]]s of 6 are 1, 2, and 3, whose sum is again 6. A pair of [[amicable number]]s 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 <math>5 \times 10^7</math> as of 1970.<ref>{{Cite journal|last=Bratley|first=Paul|last2=Lunnon|first2=Fred|last3=McKay|first3=John|date=1970|title=Amicable numbers and their distribution|url=https://www.ams.org/journals/mcom/1970-24-110/S0025-5718-1970-0271005-8/S0025-5718-1970-0271005-8.pdf|journal=Mathematics of Computation|language=en-US|volume=24|issue=110|pages=431–432|doi=10.1090/S0025-5718-1970-0271005-8|issn=0025-5718|doi-access=free}}</ref>

It is an open question whether all numbers end up at either a sociable number or at a [[Prime number|prime]] (and hence 1), or, equivalently, whether there exist numbers whose [[aliquot sequence]] never terminates, and hence grows without bound.