Define the abundancy 
of a positive integer 
as
 |
(1)
|
where 
is the divisor function. Then a pair
of distinct numbers 
is a friendly pair (and 
is said to be a friend of 
) if their abundancies are equal:
 |
(2)
|
For example, (4320, 4680) is a friendly pair since 
, 
, and
 |  |  |
(3)
|
 |  |  |
(4)
|
Another example is 
,
which has index 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.
