Sociable number

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.

Example

An example with period 4:

The sum of the proper divisors of $1264460$ ($=2^{2}\cdot 5\cdot 17\cdot 3719$) 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 $1547860$ ($=2^{2}\cdot 5\cdot 193\cdot 401$) 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 $1727636$ ($=2^{2}\cdot 521\cdot 829$) is:

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

The sum of the proper divisors of $1305184$ ($=2^{5}\cdot 40787$) is:

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

List of known sociable numbers

The following categorizes all known sociable numbers as of November 2015 by the length of the corresponding aliquot sequence:

Sequence length	Number of sequences
1 (Perfect number)	49
2 (Amicable number)	1,007,478,796[1]
4	366
5	1
6	5
8	4
9	1
28	1

Searching for sociable numbers

The aliquot sequence can be represented as a directed graph, $G_{n,s}$, for a given integer $n$, where $s(k)$ denotes the sum of the proper divisors of $k$.^[2] Cycles in $G_{n,s}$ represent sociable numbers within the interval $[1,n]$. Two special cases are loops that represent perfect numbers and cycles of length two that represent amicable pairs.

References

1. Sergei Chernykh Amicable pairs list
2. Rocha, Rodrigo Caetano; Thatte, Bhalchandra (2015), Distributed cycle detection in large-scale sparse graphs (PDF), Simpósio Brasileiro de Pesquisa Operacional (SBPO)

• 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

• A list of known sociable numbers
• Extensive tables of perfect, amicable and sociable numbers
• Weisstein, Eric W. "Sociable numbers". MathWorld.

• A003416 (smallest sociable number from each cycle) and A122726 (all sociable numbers) in OEIS