Amicable Numbers
Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number. (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. 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 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.
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).(sequence A259180 in the OEIS). (Also see 
A002025 and A002046)
A002025 and A002046)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.
As of April 2016, there are over 1,000,000,000 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.
It states that if
- p = 3×2n − 1 − 1,
- q = 3×2n − 1,
- r = 9×22n − 1 − 1,
where n > 1 is an integer and p, q, and r are prime numbers, then 2n×p×q and 2n×r are a pair of amicable numbers. This formula gives the pairs (220, 284) for n = 2, (17296, 18416) for n = 4, and (9363584, 9437056) for n = 7, but no other such pairs are known. Numbers of the form 3×2n − 1 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 n.
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.
Regular pairs
Let (m, n) be a pair of amicable numbers with m < n, and write m = gM and n = gN where g is the greatest common divisor of m and n. If M and N are both coprime to g and square free then the pair (m, n) is said to be regular (see sequence A215491 in OEIS), otherwise it is called irregular or exotic. If (m, n) is regular and M and N have i and jprime factors respectively, then (m, n) is said to be of type (i, j).
For example, with (m, n) = (220, 284), the greatest common divisor is 4 and so M = 55 and N = 71. Therefore, (220, 284) is regular of type (2, 1).
Twin amicable pairs
An amicable pair (m, n) is twin if there are no integers between m and n belonging to any other amicable pair (sequence A273259 in the OEIS).
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