sociable number

A sociable number is a generalization of the concepts of amicable numbers and perfect numbers. A set of sociable numbers is an aliquot sequence, or a sequence of numbers with each number being the sum of the factors of the preceding number, excluding the preceding number itself. In the case of sociable numbers, the sequence is cyclic (eventually returning 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, for 6 its proper divisors are 1, 2, 3 - which add to 6 itself. A pair of amicable numbers are then 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 are either sociable or end up at a prime (and hence 1), or whether conversely there exists a number whose aliquot sequence never terminates. An example with period 4:

The sum of the proper divisors of 1264460 (2² * 5 * 17 * 3719) 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 1547860 (2² * 5 * 193 * 401) 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 1727636 (2² * 521 * 829) is:

1 + 2 + 4 + 521 + 829 + 1042 + 1658 + 2084 + 3316 + 431909 + 863818 = 1305184

The sum of the proper divisors of 1305184 (2⁵ * 40787) is:

1 + 2 + 4 + 8 + 16 + 32 + 40787 + 81574 + 163148 + 326296 + 652592 = 1264460.

External links

http://xraysgi.ims.uconn.edu:8080/sociable.txt - a list of known sociable numbers compiled by David Moews
Sociable numbers, from Mathworld.

References

P. Poulet, #4865, L'intermediare des math. 25 (1918), pp. 100-101.
H. Cohen, On amicable and sociable numbers, Math. Comp. 24 (1970), pp. 423-429

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