Multiamicable Numbers

Two integers $n$ and $m<n$ are $(\alpha, \beta)$-multiamicable if

$$\sigma(m)-m=\alpha n$$

and

$$\sigma(n)-n=\beta m,$$

where $\sigma(n)$ is the Divisor Function and $\alpha,\beta$ are Positive integers. If $\alpha=\beta=1$, $(m,n)$ is an Amicable Pair.

$m$ cannot have just one distinct prime factor, and if it has precisely two prime factors, then $\alpha=1$ and $m$ is Even. Small multiamicable numbers for small $\alpha,\beta$ are given by Cohen et al. (1995). Several of these numbers are reproduced in the below table.

$\alpha$	$\beta$	$m$	$n$
1	6	76455288	183102192
1	7	52920	152280
1	7	16225560	40580280
1	7	90863136	227249568
1	7	16225560	40580280
1	7	70821324288	177124806144
1	7	199615613902848	499240550375424

See also Amicable Pair, Divisor Function

References

Cohen, G. L; Gretton, S.; and Hagis, P. Jr. ``Multiamicable Numbers.'' Math. Comput. 64, 1743-1753, 1995.