Positive Even Integers as Sum of 2 Composite Odd Integers in 2 Ways

Theorem
Let $n \in \Z_{>0}$ be a positive even integer.

Let $n$ be such that it cannot be expressed as the sum of $2$ odd positive composite integers in at least $2$ different ways.

Then $n$ belongs to the set:
 * $\left\{ {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 32, 34, 38, 40, 44, 46, 52, 56, 62, 68}\right\}$

Proof
The sequence of odd positive composite integers begins:
 * $9, 15, 21, 25, 27, 33, 35, 39, 45, 49, 51, 55, 57, 63, 65, \ldots$

which is more than we need for this proof.

Generating the possible sums of these proceeds as follows:

This exhausts all possible sums of odd positive composite integers of $68$ or less.

It remains to be demonstrated that all positive even integers over $68$ can be so expressed in $2$ different ways.

Notice that one of $n - 15, n - 25, n - 35$ is divisible by $3$.

Similarly one of $n - 9, n - 15, n - 21, n - 27, n - 33$ is divisible by $5$.

For all above $n - k$ which is composite, since $k$ is composite, $\paren {n - k} + k$ is a way to express $n$ as the sum of $2$ odd positive composite integers.

Since $n > 68$, these two ways are distinct unless $n - 15$ is divisible by $15$.

In that case, $n - 45$ is also divisible by $15$, and $n - 45 \ne 15$.

Then $n = \paren {n - 15} + 15 = \paren {n - 45} + 45$ are two ways to express $n$ as the sum of $2$ odd positive composite integers.