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'''Sociable numbers''' are numbers whose [[Aliquot_sum#Definition|aliquot sums]] form a cyclic sequence that begins and ends with the same number. They are generalizations of the concepts of [[amicable number]]s and [[perfect number]]s. The first two sociable sequences, or sociable chains, were discovered and named by the [[Belgium|Belgian]] [[mathematics|mathematician]] [[Paul Poulet (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 [[Frequency|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.
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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.
 
An example with period 4:
:The sum of the proper divisors of <math>1264460</math> (<math>=2^2\cdot5\cdot17\cdot3719</math>) 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 <math>1547860</math> (<math>=2^2\cdot5\cdot193\cdot401</math>) 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 <math>1727636</math> (<math>=2^2\cdot521\cdot829</math>) is:
::1 + 2 + 4 + 521 + 829 + 1042 + 1658 + 2084 + 3316 + 431909 + 863818 = 1305184
 
:The sum of the proper divisors of <math>1305184</math> (<math>=2^5\cdot40787</math>) is:
::1 + 2 + 4 + 8 + 16 + 32 + 40787 + 81574 + 163148 + 326296 + 652592 = 1264460.
 
The following categorizes all known sociable numbers as of October 2009 by the length of the corresponding aliquot sequence:
 
{| align="center" border="1" cellpadding="4"
|- bgcolor="#A0E0A0" align="center"
!Sequence
length
!Number of
sequences
|- align="center"
| 1
(''perfect'')
| 47
|- align="center"
| 2
(''amicable'')
|11,994,387
|- align="center"
|4
|165
|-align="center"
|5
|1
|- align="center"
|6
|5
|- align="center"
|8
|2
|- align="center"
|9
|1
|- align="center"
|28
|1
|}
 
==References==
*P. Poulet, #4865, [[L'Intermédiaire des Mathématiciens]] '''25''' (1918), pp. 100-101.
*H. Cohen, ''On amicable and sociable numbers,'' Math. Comp. '''24''' (1970), pp. 423-429
 
== External links ==
*[http://djm.cc/sociable.txt A list of known sociable numbers]
*[http://amicable.homepage.dk/tables.htm Extensive tables of perfect, amicable and sociable numbers]
*{{mathworld |urlname=SociableNumbers |title=Sociable numbers}}
 
 
{{Divisor classes}}
{{Classes of natural numbers}}
 
[[Category:Divisor function]]
[[Category:Integer sequences]]
[[Category:Number theory]]

Latest revision as of 20:13, 13 December 2014


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