Sociable numbers are generalizations of the concepts of amicable numbers and perfect numbers. A set of sociable numbers is a kind of aliquot sequence, or a sequence of numbers each of whose numbers is the sum of the factors of the preceding number, excluding the preceding number itself. For the sequence to be sociable, the sequence must be cyclic, eventually returning to its starting point. Amicable numbers are two numbers so related that the sum of the proper divisors of the one is equal to the other, unity being considered as a proper divisor but not the number itself. ... In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors, excluding itself. ... In mathematics, an aliquot sequence is a recursive sequence which can be defined in the following way: if we write σ(n) = σ1(n) to be the divisor function normally, then, the aliquot sequence of k can be written: s0 = k sn = σ(sn-1) - sn-1 or sn = σn - n For... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ...

The period of the sequence, or order of the set of sociable numbers, is the number of numbers in this cycle. A period is an arbitrary interval of time. ... Order is the opposite of anarchy and chaos. ...

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. A period is an arbitrary interval of time. ... In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors, excluding itself. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... Amicable numbers are two numbers so related that the sum of the proper divisors of the one is equal to the other, unity being considered as a proper divisor but not the number itself. ...

Excluding perfect numbers, a total of 127 sociable cycles are known.

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. In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. ... In mathematics, an aliquot sequence is a recursive sequence which can be defined in the following way: if we write σ(n) = σ1(n) to be the divisor function normally, then, the aliquot sequence of k can be written: s0 = k sn = σ(sn-1) - sn-1 or sn = σn - n For...

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

• A list of known sociable numbers
• Extensive tables of perfect, amicable and sociable numbers
• 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