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- Sociable_number abstract "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.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.An example with period 4:The sum of the proper divisors of is:1 + 2 + 4 + 5 + 10 + 17 + 20 + 34 + 68 + 85 + 170 + 340 + 3719 + 7438 + 14876 + 18595 + 37190 + 63223 + 74380 + 126446 + 252892 + 316115 + 632230 = 1547860The sum of the proper divisors of is:1 + 2 + 4 + 5 + 10 + 20 + 193 + 386 + 401 + 772 + 802 + 965 + 1604 + 1930 + 2005 + 3860 + 4010 + 8020 + 77393 + 154786 + 309572 + 386965 + 773930 = 1727636The sum of the proper divisors of is:1 + 2 + 4 + 521 + 829 + 1042 + 1658 + 2084 + 3316 + 431909 + 863818 = 1305184The sum of the proper divisors of is:1 + 2 + 4 + 8 + 16 + 32 + 40787 + 81574 + 163148 + 326296 + 652592 = 1264460.The following categorizes all known sociable numbers as of April 2013 by the length of the corresponding aliquot sequence:".
- Sociable_number wikiPageExternalLink tables.htm.
- Sociable_number wikiPageExternalLink sociable.txt.
- Sociable_number wikiPageID "332307".
- Sociable_number wikiPageRevisionID "595728056".
- Sociable_number hasPhotoCollection Sociable_number.
- Sociable_number title "Sociable numbers".
- Sociable_number urlname "SociableNumbers".
- Sociable_number subject Category:Divisor_function.
- Sociable_number subject Category:Integer_sequences.
- Sociable_number subject Category:Number_theory.
- Sociable_number type Abstraction100002137.
- Sociable_number type Arrangement107938773.
- Sociable_number type Group100031264.
- Sociable_number type IntegerSequences.
- Sociable_number type Ordering108456993.
- Sociable_number type Sequence108459252.
- Sociable_number type Series108457976.
- Sociable_number comment "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.".
- Sociable_number label "Nombre sociable".
- Sociable_number label "Numero socievole".
- Sociable_number label "Números sociables".
- Sociable_number label "Sociable number".
- Sociable_number label "相亲数链".
- Sociable_number label "社交数".
- Sociable_number sameAs Números_sociables.
- Sociable_number sameAs Nombre_sociable.
- Sociable_number sameAs Numero_socievole.
- Sociable_number sameAs 社交数.
- Sociable_number sameAs 사교수.
- Sociable_number sameAs m.01x1hw.
- Sociable_number sameAs Q1149466.
- Sociable_number sameAs Q1149466.
- Sociable_number sameAs Sociable_number.
- Sociable_number wasDerivedFrom Sociable_number?oldid=595728056.
- Sociable_number isPrimaryTopicOf Sociable_number.