Sociable number

2025-07-09T18:12:53Z

64.188.161.112: Never heard on that 'conjecture, which in unsourced anyway. Yes, for n = 3, but 3 mod 4?

{{short description|Numbers whose aliquot sums form a cyclic sequence}}
In [[mathematics]], '''sociable numbers''' are numbers whose [[Aliquot sum#Definition|aliquot sums]] form a [[periodic sequence]]. They are generalizations of the concepts of [[perfect number]]s and [[amicable number]]s. The first two sociable sequences, or sociable chains, were discovered and named by the [[Belgium|Belgian]] [[mathematician]] [[Paul Poulet (mathematician)|Paul Poulet]] in 1918.<ref>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].)</ref> 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 [[Periodic function|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 divisor]]s of 6 are 1, 2, and 3, whose sum is again 6. A pair of [[amicable number]]s 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 <math>5 \times 10^7</math> as of 1970.<ref>{{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}}</ref>

It is an open question whether all numbers end up at either a sociable number or at a [[Prime number|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 <math>1264460</math> (<math>=2^2\cdot5\cdot17\cdot3719</math>) 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 <math>1547860</math> (<math>=2^2\cdot5\cdot193\cdot401</math>) 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 <math>1727636</math> (<math>=2^2\cdot521\cdot829</math>) is
::1 + 2 + 4 + 521 + 829 + 1042 + 1658 + 2084 + 3316 + 431909 + 863818 = 1305184, and

:the sum of the proper divisors of <math>1305184</math> (<math>=2^5\cdot40787</math>) 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:

{| align="center" border="1" cellpadding="4"
|- bgcolor="#A0E0A0" align="center"
!Sequence
length
!Number of known
sequences
! lowest number
in sequence<ref>https://oeis.org/A003416 cross referenced with https://oeis.org/A052470</ref>
|- align="center"
|1
(''[[Perfect number]]'')
|52
|6
|- align="center"
|2
(''[[Amicable number]]'')
| 1 billion+<ref>Sergei Chernykh: [http://sech.me/ap/ Amicable pairs list]</ref>
|220
|- align="center"
|4
|5398
| 1,264,460
|- align="center"
|5
|1
|12,496
|- align="center"
|6
|5
|21,548,919,483
|- align="center"
|8
|4
|1,095,447,416
|- align="center"
|9
|1
|805,984,760
|- align="center"
|28
|1
|14,316
|}

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]], <math>G_{n,s}</math>, for a given integer <math>n</math>, where <math>s(k)</math> denotes the
sum of the proper divisors of <math>k</math>.<ref>{{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}}</ref>
[[cycle (graph theory)|Cycles]] in <math>G_{n,s}</math> represent sociable numbers within the interval <math>[1,n]</math>. 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==
{{Reflist}}
*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}}
*[[oeis:A003416|A003416 (smallest sociable number from each cycle)]] and [[oeis:A122726|A122726 (all sociable numbers)]] in [[OEIS]]

{{Divisor classes}}
{{Classes of natural numbers}}

[[Category:Arithmetic dynamics]]
[[Category:Divisor function]]
[[Category:Integer sequences]]
[[Category:Number theory]]

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Sociable number