# Amicable numbers

### 1727

For example, 1264460 \mapsto 1547860 \mapsto 1727636 \mapsto 1305184 \mapsto 1264460 \mapsto\dots are sociable numbers of order 4. ==== Searching for sociable numbers ==== The aliquot sequence can be represented as a directed graph, G_{n,s}, for a given integer n, where s(k) denotes the sum of the proper divisors of k. Cycles in G_{n,s} represent sociable numbers within the interval [1,n].

### 1750

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 .

### 1841

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).

### 1866

The second smallest pair, (1184, 1210), was discovered in 1866 by a then teenage B.

### 1946

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.

### 1972

It was extended further by Borho in 1972.

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