A pair of positive integers 
such that the equations
 |
(1)
|
have a positive integer solution 
, where 
is the divisor function. If 
is prime, then 
is an amicable pair
(te Riele 1986). 
is a "special" breeder if
 |  |  |
(2)
|
 |  |  |
(3)
|
where 
and 
are relatively prime,
. If regular amicable pairs of
type 
with 
are of the form 
with 
prime, then 
are special breeders (te Riele 1986).
."
Math. Comput. 47, 361-368 and S9-S35, 1986.