Breeder

A pair of Positive Integers $(a_1, a_2)$ such that the equations

$$a_1+a_2x=\sigma(a_1)=\sigma(a_2)(x+1)$$

have a Positive Integer solution $x$, where $\sigma(n)$ is the Divisor Function. If $x$ is Prime, then $(a_1, a_2x)$ is an Amicable Pair (te Riele 1986). $(a_1, a_2)$ is a ``special'' breeder if

$$a_1 = au$$

$$a_2 = a,$$

where $a$ and $u$ are Relatively Prime, $(a,u)=1$. If regular amicable pairs of type $(i,1)$ with $i\geq 2$ are of the form $(au,ap)$ with $p$ Prime, then $(au,a)$ are special breeders (te Riele 1986).

References

te Riele, H. J. J. ``Computation of All the Amicable Pairs Below $10^{10}$.'' Math. Comput. 47, 361-368 and S9-S35, 1986.