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.[1] 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, and searches for them have been made up to ${\displaystyle 5\times 10^{7}}$ as of 1970 [2].

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 ${\displaystyle 1264460}$ (${\displaystyle =2^{2}\cdot 5\cdot 17\cdot 3719}$) 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 ${\displaystyle 1547860}$ (${\displaystyle =2^{2}\cdot 5\cdot 193\cdot 401}$) 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 ${\displaystyle 1727636}$ (${\displaystyle =2^{2}\cdot 521\cdot 829}$) is:
1 + 2 + 4 + 521 + 829 + 1042 + 1658 + 2084 + 3316 + 431909 + 863818 = 1305184
The sum of the proper divisors of ${\displaystyle 1305184}$ (${\displaystyle =2^{5}\cdot 40787}$) 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 July 2018 by the length of the corresponding aliquot sequence:

Sequence

length

Number of known

sequences

1 50
2 1222207191[3]
4 5398
5 1
6 5
8 4
9 1
28 1

It is conjectured that if n = 3 mod 4, then there are no such sequence with length n.

The smallest number of the only known 28-cycle is 14316.

Searching for sociable numbers

The aliquot sequence can be represented as a directed graph, ${\displaystyle G_{n,s}}$, for a given integer ${\displaystyle n}$, where ${\displaystyle s(k)}$ denotes the sum of the proper divisors of ${\displaystyle k}$.[4] Cycles in ${\displaystyle G_{n,s}}$ represent sociable numbers within the interval ${\displaystyle [1,n]}$. Two special cases are loops that represent perfect numbers and cycles of length two that represent amicable pairs.

Conjecture of the sum of sociable number cycles

As the number of sociable number cycles with length greater than 2 approaches infinity, the percentage of the sums of the sociable number cycles divisible by 10 approaches 100%.(sequence A292217 in the OEIS).

References

1. ^ P. Poulet, #4865, L'Intermédiaire des Mathématiciens 25 (1918), pp. 100–101. (The full text can be found at ProofWiki: Catalan-Dickson Conjecture.)
2. ^ Bratley, Paul; Lunnon, Fred; McKay, John (1970). "Amicable numbers and their distribution". Mathematics of Computation. 24 (110): 431–432. doi:10.1090/S0025-5718-1970-0271005-8. ISSN 0025-5718.
3. ^ Sergei Chernykh Amicable pairs list
4. ^ Rocha, Rodrigo Caetano; Thatte, Bhalchandra (2015), Distributed cycle detection in large-scale sparse graphs (PDF), Simpósio Brasileiro de Pesquisa Operacional (SBPO)
• H. Cohen, On amicable and sociable numbers, Math. Comp. 24 (1970), pp. 423–429