Sociable number

{{short description|Numbers whose aliquot sums form a cyclic sequence}}

In mathematics, sociable numbers are numbers whose aliquot sums form a periodic sequence. They are generalizations of the concepts of perfect numbers and amicable numbers. The first two sociable sequences, or sociable chains, were discovered and named by the Belgian mathematician Paul Poulet in 1918.P. Poulet, #4865, L'Intermédiaire des Mathématiciens 25 (1918), pp. 100–101. (The full text can be found at [https://proofwiki.org/wiki/Catalan-Dickson_Conjecture ProofWiki: Catalan-Dickson Conjecture].) In a sociable sequence, each number is the sum of the proper divisors 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.{{Cite journal|last=Bratley|first=Paul|last2=Lunnon|first2=Fred|last3=McKay|first3=John|date=1970|title=Amicable numbers and their distribution|url=https://www.ams.org/journals/mcom/1970-24-110/S0025-5718-1970-0271005-8/S0025-5718-1970-0271005-8.pdf|journal=Mathematics of Computation|language=en-US|volume=24|issue=110|pages=431–432|doi=10.1090/S0025-5718-1970-0271005-8|issn=0025-5718|doi-access=free}}

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

As an example, the number 1,264,460 is a sociable number whose cyclic aliquot sequence has a period of 4:

:The sum of the proper divisors of $1264460$ ( $=2^2\cdot5\cdot17\cdot3719$ ) 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 $1547860$ ( $=2^2\cdot5\cdot193\cdot401$ ) 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 $1727636$ ( $=2^2\cdot521\cdot829$ ) is

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

:the sum of the proper divisors of $1305184$ ( $=2^5\cdot40787$ ) 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|2024|10|lc=y}} by the length of the corresponding aliquot sequence:

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bgcolor="#A0E0A0" align="center"

!Sequence

length

!Number of known

sequences

! lowest number

in sequencehttps://oeis.org/A003416 cross referenced with https://oeis.org/A052470

align="center"

(Perfect number)

|52

align="center"

(Amicable number)

| 1 billion+Sergei Chernykh: [http://sech.me/ap/ Amicable pairs list]

|220

align="center"

|5398

| 1,264,460

align="center"

|12,496

align="center"

|21,548,919,483

align="center"

|1,095,447,416

align="center"

|805,984,760

align="center"

|28

|14,316

It is conjectured that if n is congruent to 3 modulo 4 then there is 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 {{OEIS|A072890}}.

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 $k$ .{{citation|title=Distributed cycle detection in large-scale sparse graphs|first1=Rodrigo Caetano|last1=Rocha|first2=Bhalchandra|last2=Thatte|year=2015|publisher=Simpósio Brasileiro de Pesquisa Operacional (SBPO)|doi=10.13140/RG.2.1.1233.8640}}

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 proportion of the sums of the sociable number cycles divisible by 10 approaches 1 {{OEIS|A292217}}.

References

H. Cohen, On amicable and sociable numbers, Math. Comp. 24 (1970), pp. 423–429

External links

[http://djm.cc/sociable.txt A list of known sociable numbers]
[https://web.archive.org/web/20140502102524/http://amicable.homepage.dk/tables.htm Extensive tables of perfect, amicable and sociable numbers]
{{mathworld |urlname=SociableNumbers |title=Sociable numbers}}
A003416 (smallest sociable number from each cycle) and A122726 (all sociable numbers) in OEIS

Category:Arithmetic dynamics

Category:Divisor function

Category:Integer sequences

Category:Number theory