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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. | | Eusebio Stanfill is what's developed on my birth records although it is not the name on my birth certificate. Vermont is just where my home is. Software developing has been my day job for a while. To bake is the only sport my wife doesn't approve of. You can acquire my website here: http://prometeu.net<br><br>Also visit my page ... clash of clans cheat - [http://prometeu.net prometeu.net], |
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| The [[Frequency|period]] of the sequence, or order of the set of sociable numbers, is the number of numbers in this cycle.
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| 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.
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| An example with period 4:
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| :The sum of the proper divisors of <math>1264460</math> (<math>=2^2\cdot5\cdot17\cdot3719</math>) is:
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| ::1 + 2 + 4 + 5 + 10 + 17 + 20 + 34 + 68 + 85 + 170 + 340 + 3719 + 7438 + 14876 + 18595 + 37190 + 63223 + 74380 + 126446 + 252892 + 316115 + 632230 = 1547860 | |
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| :The sum of the proper divisors of <math>1547860</math> (<math>=2^2\cdot5\cdot193\cdot401</math>) is:
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| ::1 + 2 + 4 + 5 + 10 + 20 + 193 + 386 + 401 + 772 + 802 + 965 + 1604 + 1930 + 2005 + 3860 + 4010 + 8020 + 77393 + 154786 + 309572 + 386965 + 773930 = 1727636
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| :The sum of the proper divisors of <math>1727636</math> (<math>=2^2\cdot521\cdot829</math>) is:
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| ::1 + 2 + 4 + 521 + 829 + 1042 + 1658 + 2084 + 3316 + 431909 + 863818 = 1305184
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| :The sum of the proper divisors of <math>1305184</math> (<math>=2^5\cdot40787</math>) is:
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| ::1 + 2 + 4 + 8 + 16 + 32 + 40787 + 81574 + 163148 + 326296 + 652592 = 1264460.
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| The following categorizes all known sociable numbers as of October 2009 by the length of the corresponding aliquot sequence:
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| {| align="center" border="1" cellpadding="4"
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| |- bgcolor="#A0E0A0" align="center"
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| !Sequence
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| length
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| !Number of
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| sequences
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| |- align="center"
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| | 1
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| (''perfect'')
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| | 47
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| |- align="center"
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| | 2
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| (''amicable'')
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| |11,994,387
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| |- align="center"
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| |4
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| |165
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| |-align="center"
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| |5
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| |1
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| |- align="center"
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| |6
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| |5
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| |- align="center"
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| |8
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| |2
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| |- align="center"
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| |9
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| |1
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| |- align="center"
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| |28
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| |1
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| |}
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| ==References==
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| *P. Poulet, #4865, [[L'Intermédiaire des Mathématiciens]] '''25''' (1918), pp. 100-101.
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| *H. Cohen, ''On amicable and sociable numbers,'' Math. Comp. '''24''' (1970), pp. 423-429
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| == External links ==
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| *[http://djm.cc/sociable.txt A list of known sociable numbers]
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| *[http://amicable.homepage.dk/tables.htm Extensive tables of perfect, amicable and sociable numbers]
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| *{{mathworld |urlname=SociableNumbers |title=Sociable numbers}}
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| {{Divisor classes}}
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| {{Classes of natural numbers}}
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| [[Category:Divisor function]]
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| [[Category:Integer sequences]]
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| [[Category:Number theory]]
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