Definition:Sociable Chain

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Definition

Let $m$ be a positive integer.

Let $\map s m$ be the aliquot sum of $m$.


Define the sequence $\sequence {a_k}$ recursively as:

$a_{k + 1} = \begin{cases} m & : k = 0 \\ \map s {a_k} & : k > 0 \end{cases}$


A sociable chain is such a sequence $\sequence {a_k}$ where:

$a_r = a_0$

for some $r > 0$.


Order of Sociable Chain

The order of $a_k$ is the smallest $r \in \Z_{>0}$ such that

$a_r = a_0$


Examples

Sociable Chain starting at $12 \, 496$

The sociable chain whose smallest element is $12 \, 496$ is of order $5$.

It goes:

$12 \, 496 \to 14 \, 288 \to 15 \, 472 \to 14 \, 536 \to 14 \, 264 \to 12 \, 496$


Sociable Chain starting at $14 \, 316$

The longest known sociable chain, at time of writing ($9$th March $2017$), is of order $28$.

Its smallest element is $14 \, 316$.


Also known as

A sociable chain is known in some sources as a sociable cycle.

Beware of the temptation to refer to it as a social chain -- this term is not used in mathematics.


A sociable chain of order $2$ is generally known as an amicable pair.

A sociable chain of order $3$ is also known by some mathematicians as a crowd, but it is unknown whether any exist.


Also see

  • Results about sociable numbers can be found here.


Historical Note

The concept of a sociable chain was discussed by Paul Poulet, who wrote a paper on the subject.


The next sociable chains to be discovered were found in $1969$ and published in $1970$ by Henri Cohen, who found $9$ of them, all of order $4$.

Several more have been found since, by various people, including David Moews, Paul C. Moews, Achim Flammenkamp and Ren Yuanhua.

Most of these are of order $4$, but two are of order $8$ and one of order $9$

Even more recently, $5$ sociable chains of order $6$ have been discovered.


Full Text of the Article by Poulet

Si l'on considère un nombre entier a, la somme b de ses parties aliquotes, la somme c des parties aliquotes de b, la somme d des parties aliquotes de c et ainsi de suite, on obtient un développement qui, poussé indéfiniment, peut se présenter sous trois aspects différents:

  1. Le plus souvent on finit par tomber sur un nombre premier, puis sur l'unité. Le développement est fini.
  2. On retrouve à un moment donné un nombre déjà rencontré. Le développement est indéfini et périodique. Si la période n'a qu'un terme, ce terme est un nombre parfait. Si la période a deux termes, ces termes sont des nombres amiables. La période peut avoir plus de deux termes, qu'on pourrait appeler, pour garder la même terminologie, des nombres sociables. Par exemple le nombre $12496$ engendre une période de $4$ termes, le nombre $14316$ une période de $28$ termes.
  3. Enfin dans certains cas, on arrive à des nombres très grands qui rendent le calcul insupportable. Exemple: le nombre $138$.

Cela étant, je demande:

  1. Si ce troisième cas existe réellement ou si, en poursuivant indéfiniment le calcul, il ne se résoudrait pas nécessairement dans l'un ou l'autre des deux premiers, comme je suis porté à le croire.
  2. Si l'on connaît d'autres groupes sociables que ceux donnés plus haut, notamment des groupes de trois termes. (Il est inutile, je pense, d'essayer les nombres inférieurs à 12000 que j'ai tous examinés.)


This can be rendered in English as:

If one considers a whole number $a$, the sum $b$ of its proper divisors, the sum $c$ of the proper divisors of $b$, the sum $d$ of the proper divisors of $c$, and so on, one creates a sequence that, continued indefinitely, can develop in three ways:

  1. The most frequent is to arrive at a prime number, then at unity. The sequence ends here.
  2. One arrives at a previously calculated number. The sequence is indefinite and periodic. If the period is one, the number is perfect. If the period is two, the numbers are amicable. But the period can be longer than two, involving what I will call, to keep the same terminology, sociable numbers. For example, the number $12496$ creates a period of four terms, the number $14316$ a period of $28$ terms.
  3. Finally, in some cases a sequence creates very large numbers that become impossible to resolve into divisors. For example, the number $138$.

This being so, I ask:

  1. If this third case really exists or if, calculating long enough, one would not necessarily end in one of the two other cases, as I am driven to believe.
  2. If sociable chains other than those above can be found, especially chains of three terms. (It will be pointless, I think, to try numbers below $12000$, because I have tested all of them.)


Sources