Definition:Amicable Pair

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Definition

Let $m \in \Z_{>0}$ and $n \in \Z_{>0}$ be (strictly) positive integers.


Definition 1

$m$ and $n$ are an amicable pair if and only if:

the aliquot sum of $m$ is equal to $n$

and:

the aliquot sum of $n$ is equal to $m$.


Definition 2

$m$ and $n$ are an amicable pair if and only if:

$\sigma \left({m}\right) = \sigma \left({n}\right) = m + n$

where $\sigma \left({m}\right)$ denotes the $\sigma$ function.


Definition 3

$m$ and $n$ are an amicable pair if and only if they form a sociable chain of order $2$.


Sequence

The sequence of amicable pairs begins:

$\left({220, 284}\right), \left({1184, 1210}\right), \left({2620, 2924}\right), \left({5020, 5564}\right), \left({6232, 6368}\right), \left({10744, 10856}\right), \left({12285, 14595}\right), \left({17296, 18416}\right), \left({63020, 76084}\right)$

This sequence is A259180 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

The sequence of the $1$st elements is A002025 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

The sequence of the $2$nd elements is A002046 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Examples

$220$ and $284$

$220$ and $284$ are the smallest amicable pair:

$\map \sigma {220} = \map \sigma {284} = 504 = 220 + 284$


$1184$ and $1210$

$1184$ and $1210$ are the $2$nd amicable pair:

$\map \sigma {1184} = \map \sigma {1210} = 2394 = 1184 + 1210$


$2620$ and $2924$

$2620$ and $2924$ are the $3$rd amicable pair:

$\map \sigma {2620} = \map \sigma {2924} = 5544 = 2620 + 2924$


$5020$ and $5564$

$5020$ and $5564$ are the $4$th amicable pair:

$\map \sigma {5020} = \map \sigma {5564} = 10 \, 584 = 5020 + 5564$


$6232$ and $6368$

$6232$ and $6368$ are the $5$th amicable pair:

$\sigma \left({6232}\right) = \sigma \left({6368}\right) = 12 \, 600 = 6232 + 6368$


$10 \, 744$ and $10 \, 856$

$10 \, 744$ and $10 \, 856$ are the $6$th amicable pair:

$\sigma \left({10 \, 744}\right) = \sigma \left({10 \, 856}\right) = 21 \, 600 = 10 \, 744 + 10 \, 856$


$12 \, 285$ and $14 \, 595$

$12 \, 285$ and $14 \, 595$ are the $7$th amicable pair and the smallest odd amicable pair:

$\map \sigma {12 \, 285} = \map \sigma {14 \, 595} = 26 \, 880 = 12 \, 285 + 14 \, 595$


$17 \, 296$ and $18 \, 416$

$17 \, 296$ and $18 \, 416$ are the $8$th amicable pair:

$\map \sigma {17 \, 296} = \map \sigma {18 \, 416} = 35 \, 712 = 17 \, 296 + 18 \, 416$


Also known as

Some sources refer to them as friendly pairs, but this is deprecated as the term friendly pair is usually used for something different.

Many sources refer to the elements of an amicable pair as amicable numbers. While this usage is common and widespread, it is preferred not to be used on $\mathsf{Pr} \infty \mathsf{fWiki}$ in order that the distinction is clear between amicable pairs, amicable triplets and so on.


Also see

  • Results about amicable pairs can be found here.


Historical Note

The first mathematician to explore amicable pairs systematically was Leonhard Paul Euler.

He published a list of $64$ examples.

There are now over $40 \, 000$ amicable pairs now known, including all such pairs where the smaller pair is under $1 \, 000 \, 000$.

Techniques for generating new amicable pairs from existing ones strongly suggest that there is an infinite number of them, but this still has to be rigorously proven.


Sources