Editing Amicable numbers (section)

== Generalizations ==

=== Amicable tuples ===
Amicable numbers <math>(m, n)</math> satisfy <math>\sigma(m)-m=n</math> and <math>\sigma(n)-n=m</math> which can be written together as <math>\sigma(m)=\sigma(n)=m+n</math>. This can be generalized to larger tuples, say <math>(n_1,n_2,\ldots,n_k)</math>, where we require

:<math>\sigma(n_1)=\sigma(n_2)= \dots =\sigma(n_k) = n_1+n_2+ \dots +n_k</math>

For example, (1980, 2016, 2556) is an [[amicable triple]] {{OEIS|id=A125490}}, and (3270960, 3361680, 3461040, 3834000) is an amicable quadruple {{OEIS|id=A036471}}.

Amicable [[multiset]]s are defined analogously and generalizes this a bit further {{OEIS|id=A259307}}.

=== Sociable numbers ===
{{main article|Sociable number}}

Sociable numbers are the numbers in cyclic lists of numbers (with a length greater than 2) where each number is the sum of the proper divisors of the preceding number. For example, <math>1264460 \mapsto 1547860 \mapsto 1727636 \mapsto 1305184 \mapsto 1264460 \mapsto\dots</math> are sociable numbers of order 4.

==== 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]].