Smallest Pair of Quasiamicable Numbers

Theorem
The smallest pair of quasiamicable numbers is $48$ and $75$.

Proof
From Quasiamicable Numbers: $48$ and $75$ we have that $48$ and $75$ are quasiamicable numbers.

It remains to be demonstrated that these are indeed the smallest such.

Let $n$ be the smaller number of the quasiamicable pair.

Then we must have $\map {\sigma_1} n - n - 1 > n$.

Since $\map {\sigma_1} n > 2 n$, $n$ is abundant.

The abundant numbers for $n < 48$ are:
 * $12, 18, 20, 24, 30, 36, 40, 42$

And we have:

hence $n$ is at least $48$, proving the result.