Even Integers not Sum of Two Abundant Numbers

Theorem
The even integers which are not the sum of $2$ abundant numbers are:
 * All even integers less than $24$;
 * $26, 28, 34, 46$

Proof
From Sequence of Abundant Numbers, the first few abundant numbers are:
 * $12, 18, 20, 24, 30, 36, 40, 42, 48$

Immediately we see that any number less than $2 \times 12 = 24$ cannot be expressed as a sum of $2$ abundant numbers.

Next sum of $2$ abundant numbers is $12 + 18 = 30$, so $26$ and $28$ are not sums of $2$ abundant numbers.

Since none of the differences above are abundant numbers, $34$ and $46$ are not sums of $2$ abundant numbers.

We demonstrate that $32$ and all even numbers from $36$ to $66$ except $46$ are sums of $2$ abundant numbers:

The numbers $36, 42, 48, 54, 60$ and $66$ are multiples of $6$.

By Multiple of Perfect Number is Abundant, any multiple of $6$ greater than $6$ is abundant.

Hence these numbers can be expressed as:
 * $12 + \paren {n - 12}$

which are sums of $2$ multiples of $6$ greater than $6$.

Now that we show that all even numbers greater than $66$ are sums of $2$ abundant numbers.

By Multiple of Abundant Number is Abundant, any multiple of $20$ is abundant.

By Largest Number not Expressible as Sum of Multiples of Coprime Integers/Generalization, any even number greater than:
 * $\dfrac {6 \times 20} {\gcd \set {6, 20}} - 6 - 20 = 34$

is a sum of (possibly zero) multiples of $6$ and $20$.

Hence any even number greater than:
 * $34 + 6 \times 2 + 20 = 66$

is a sum of a multiple of $6$ greater than $6$ and a non-zero multiple of $20$, which by above is a sum of $2$ abundant numbers.

This shows that the list above is complete.

Also see

 * Largest Integer not Sum of Two Abundant Numbers