Sociable number

In mathematics, 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, and searches for them have been made up to $5\times 10^{7}$ as of 1970.

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, and

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 sociable numbers (cycle length ≥3) are

12496, 14264, 14288, 14316, 14536, 15472, 17716, 19116, 19916, 22744, 22976, 31704, 45946, 47616, 48976, 83328, 97946, 122410, 152990, 177792, 243760, 274924, 275444, 285778, 294896, 295488, … (sequence A122726 in the OEIS)

The sociable numbers (include cycle length 1 and 2, i.e. include perfect numbers and amicable numbers) are

6, 28, 220, 284, 496, 1184, 1210, 2620, 2924, 5020, 5564, 6232, 6368, 8128, 10744, 10856, 12285, 12496, 14264, 14288, 14316, 14536, 14595, 15472, 17296, 17716, 18416, 19116, 19916, 22744, 22976, 31704, 45946, 47616, 48976, … (sequence A347770 in the OEIS)

Smallest member of the sociable cycles with length ≥3 are

12496, 14316, 1264460, 2115324, 2784580, 4938136, 7169104, 18048976, 18656380, 28158165, 46722700, 81128632, 174277820, 209524210, 330003580, 498215416, 805984760, … (sequence A003416 in the OEIS)

Cycle length ≥3 of sociable numbers are

5, 28, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 9, 8, 4, 8, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6, 4, 4, 4, 4, 4, 6, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6, 4, 4, 4, … (sequence A052470 in the OEIS)

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	lowest number in sequence
1 (Perfect number)	51	6
2 (Amicable number)	1225736919	220
4	5398	1,264,460
5	1	12,496
6	5	21,548,919,483
8	4	1,095,447,416
9	1	805,984,760
28	1	14,316

It is conjectured that if n is congruent to 3 modulo 4 then there are no such sequence with length n.

The 5-cycle sequence is: 12496, 14288, 15472, 14536, 14264

The only known 28-cycle is: 14316, 19116, 31704, 47616, 83328, 177792, 295488, 629072, 589786, 294896, 358336, 418904, 366556, 274924, 275444, 243760, 376736, 381028, 285778, 152990, 122410, 97946, 48976, 45946, 22976, 22744, 19916, 17716. (sequence A072890 in the OEIS).

These two sequences provide the only sociable numbers below 1 million (other than the perfect and amicable numbers).

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 $k$. 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.

Conjecture of the sum of sociable number cycles

It is conjectured that 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

• 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