A pair of positive integers (a_1, a_2) such that the equations a_1 + a_2 x = σ(a_1) = σ(a_2)(x + 1) have a positive integer solution x, where σ(n) is the divisor function. If x is prime, then (a_1, a_2 x) is an amicable pair (te Riele 1986). (a_1, a_2) is a "special" breeder if a_1 | = | a u a_2 | = | a, where a and u are relatively prime, (a, u) = 1. If regular amicable pairs of type (i, 1) with i>=2 are of the form (a u, a p) with p prime, then (a u, a) are special breeders (te Riele 1986).
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