A friendly pair is a pair of distinct positive integers 
having equal abundancies,
 |
(1)
|
Here, the abundancy 
of a positive integer 
is defined as
 |
(2)
|
and 
is the divisor
function. In a friendly pair, 
is said to be a friend of 
. For example, (4320, 4680) is a friendly pair since
, 
, and
 |  |  |
(3)
|
 |  |  |
(4)
|
Another example is 
, which has abundancy
5/2. The first few friendly pairs, ordered by smallest maximum element are (6, 28),
(30, 140), (80, 200), (40, 224), (12, 234), (84, 270), (66, 308), ... (OEIS A050972
and A050973).
Friendly triples and higher-order tuples are also possible. Friendly triples include (2160, 5400, 13104), (9360, 21600, 23400), and (4320, 4680, 26208), friendly quadruples include (6, 28, 496, 8128), (3612, 11610, 63984, 70434), (3948, 12690, 69936, 76986), and friendly quintuples include (84, 270, 1488, 1638, 24384), (30, 140, 2480, 6200, 40640), (420, 7440, 8190, 18600, 121920).
Numbers that have friends are called friendly numbers, and numbers that do not have friends are
called solitary numbers. A sufficient (but not
necessary) condition for 
to be a solitary number is that 
, where 
is the greatest
common divisor of 
and 
. There are some numbers that can easily be proved to
be solitary, but the status of numbers 10, 14,
15, 20, and many others remains unknown (Hickerson 2002).
Hoffman (1998, p. 45) uses the term "friendly numbers" to describe amicable pairs.
