Amicable numbers

Amicable numbers are two different numbers related in such a way that the sum of the proper divisors of each is equal to the other number.
The smallest pair of amicable numbers is (220, 284). They are amicable because the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284; and the proper divisors of 284 are 1, 2, 4, 71 and 142, of which the sum is 220. (A proper divisor of a number is a positive factor of that number other than the number itself. For example, the proper divisors of 6 are 1, 2, and 3.) 
A pair of amicable numbers constitutes an aliquot sequence of period 2. It is unknown if there are infinitely many pairs of amicable numbers.
A related concept is that of a perfect number, which is a number that equals the sum of "its own" proper divisors, in other words a number which forms an aliquot sequence of period 1. Numbers that are members of an aliquot sequence with period greater than 2 are known as sociable numbers.
The first ten amicable pairs are: (220, 284), (1184, 1210), (2620, 2924), (5020, 5564), (6232, 6368), (10744, 10856), (12285, 14595), (17296, 18416), (63020, 76084), and (66928, 66992). . (Also see and )
History.
Amicable numbers were known to the Pythagoreans, who credited them with many mystical properties. A general formula by which some of these numbers could be derived was invented circa 850 by the Iraqi mathematician Thābit ibn Qurra (826–901). Other Arab mathematicians who studied amicable numbers are al-Majriti (died 1007), al-Baghdadi (980–1037), and al-Fārisī (1260–1320). The Iranian mathematician Muhammad Baqir Yazdi (16th century) discovered the pair (9363584, 9437056), though this has often been attributed to Descartes. Much of the work of Eastern mathematicians in this area has been forgotten.
Thābit ibn Qurra's formula was rediscovered by Fermat (1601–1665) and Descartes (1596–1650), to whom it is sometimes ascribed, and extended by Euler (1707–1783). It was extended further by Borho in 1972. Fermat and Descartes also rediscovered pairs of amicable numbers known to Arab mathematicians. Euler also discovered dozens of new pairs. The second smallest pair, (1184, 1210), was discovered in 1866 by a then teenage B. Nicolò I. Paganini (not to be confused with the composer and violinist), having been overlooked by earlier mathematicians.
By 1946 there were 390 known pairs, but the advent of computers has allowed the discovery of many thousands since then. Exhaustive searches have been carried out to find all pairs less than a given bound, this bound being extended from 108 in 1970, to 1010 in 1986, 1011 in 1993, 1017 in 2015, and to 1018 in 2016.
, there are over 1,226,910,693 known amicable pairs.
Rules for generation.
While these rules do generate some pairs of amicable numbers, many other pairs are known, so these rules are by no means comprehensive.
In particular, the two rules below produce only even amicable pairs, so they are of no interest for the open problem of finding amicable pairs coprime to 210 = 2·3·5·7, while over 1000 pairs coprime to 30 = 2·3·5 are known [García, Pedersen & te Riele (2003), Sándor & Crstici (2004)]. 
Thābit ibn Qurra theorem.
The Thābit ibn Qurra theorem is a method for discovering amicable numbers invented in the ninth century by the Arab mathematician Thābit ibn Qurra.
It states that if
where is an integer and , , and are prime numbers, then and are a pair of amicable numbers. This formula gives the pairs for , for , and for , but no other such pairs are known. Numbers of the form are known as Thabit numbers. In order for Ibn Qurra's formula to produce an amicable pair, two consecutive Thabit numbers must be prime; this severely restricts the possible values of .
To establish the theorem, Thâbit ibn Qurra proved nine lemmas divided into two groups. The first three lemmas deal with the determination of the aliquot parts of a natural integer. The second group of lemmas deals more specifically with the formation of perfect, abundant and deficient numbers.
Euler's rule.
"Euler's rule" is a generalization of the Thâbit ibn Qurra theorem. It states that if
where are integers and , , and are prime numbers, then and are a pair of amicable numbers. Thābit ibn Qurra's theorem corresponds to the case . Euler's rule creates additional amicable pairs for with no others being known. Euler (1747 & 1750) overall found 58 new pairs to make all the by then existing pairs into 61.
Regular pairs.
Let (, ) be a pair of amicable numbers with , and write and where is the greatest common divisor of and . If and are both coprime to and square free then the pair (, ) is said to be regular ; otherwise, it is called irregular or exotic. If (, ) is regular and and have and prime factors respectively, then is said to be of type .
For example, with , the greatest common divisor is and so and . Therefore, is regular of type .
Twin amicable pairs.
An amicable pair is twin if there are no integers between and belonging to any other amicable pair 
Other results.
In every known case, the numbers of a pair are either both even or both odd. It is not known whether an even-odd pair of amicable numbers exists, but if it does, the even number must either be a square number or twice one, and the odd number must be a square number. However, amicable numbers where the two members have different smallest prime factors do exist: there are seven such pairs known. Also, every known pair shares at least one common prime factor. It is not known whether a pair of coprime amicable numbers exists, though if any does, the product of the two must be greater than 1067. Also, a pair of coprime amicable numbers cannot be generated by Thabit's formula (above), nor by any similar formula.
In 1955, Paul Erdős showed that the density of amicable numbers, relative to the positive integers, was 0.
In 1968, Martin Gardner noted that most even amicable pairs known at his time have sums divisible by 9, and a rule for characterizing the exceptions was obtained. 
According to the sum of amicable pairs conjecture, as the number of the amicable numbers approaches infinity, the percentage of the sums of the amicable pairs divisible by ten approaches 100% .
Generalizations.
Amicable tuples.
Amicable numbers formula_1 satisfy formula_2 and formula_3 which can be written together as formula_4. This can be generalized to larger tuples, say formula_5, where we require
For example, (1980, 2016, 2556) is an amicable triple , and (3270960, 3361680, 3461040, 3834000) is an amicable quadruple .
Amicable multisets are defined analogously and generalizes this a bit further .
Sociable numbers.
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, formula_7 are sociable numbers of order 4.
Searching for sociable numbers.
The aliquot sequence can be represented as a directed graph, formula_8, for a given integer formula_9, where formula_10 denotes the
sum of the proper divisors of formula_11.
Cycles in formula_8 represent sociable numbers within the interval formula_13. Two special cases are loops that represent perfect numbers and cycles of length two that represent amicable pairs.