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

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:
:$\left\{ {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 32, 34, 38, 40, 44, 46, 52, 56, 62, 68}\right\}$
{{OEIS|A284788}}

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.
{{OEIS|A071904}}
Generating the possible sums of these proceeds as follows:
{{begin-eqn}}
{{eqn | l = 18
      | r = 9 + 9
      | c = 
}}
{{eqn | l = 24
      | r = 9 + 15
      | c = 
}}
{{eqn | l = 30
      | r = 9 + 21
      | c = 
}}
{{eqn | r = 15 + 15
      | c = 
}}
{{eqn | l = 34
      | r = 9 + 25
      | c = 
}}
{{eqn | l = 36
      | r = 9 + 27
      | c = 
}}
{{eqn | r = 15 + 21
      | c = 
}}
{{eqn | l = 40
      | r = 15 + 25
      | c = 
}}
{{eqn | l = 42
      | r = 9 + 33
      | c = 
}}
{{eqn | r = 15 + 27
      | c = 
}}
{{eqn | r = 21 + 21
      | c = 
}}
{{eqn | l = 44
      | r = 9 + 35
      | c = 
}}
{{eqn | l = 46
      | r = 21 + 25
      | c = 
}}
{{eqn | l = 48
      | r = 9 + 39
      | c = 
}}
{{eqn | r = 15 + 33
      | c = 
}}
{{eqn | r = 21 + 27
      | c = 
}}
{{eqn | l = 50
      | r = 15 + 35
      | c = 
}}
{{eqn | r = 25 + 25
      | c = 
}}
{{eqn | l = 52
      | r = 25 + 27
      | c = 
}}
{{eqn | l = 54
      | r = 9 + 45
      | c = 
}}
{{eqn | r = 15 + 39
      | c = 
}}
{{eqn | r = 21 + 33
      | c = 
}}
{{eqn | r = 27 + 27
      | c = 
}}
{{eqn | l = 56
      | r = 21 + 35
      | c = 
}}
{{eqn | l = 58
      | r = 9 + 49
      | c = 
}}
{{eqn | r = 25 + 33
      | c = 
}}
{{eqn | l = 60
      | r = 9 + 51
      | c = 
}}
{{eqn | r = 15 + 45
      | c = 
}}
{{eqn | r = 21 + 39
      | c = 
}}
{{eqn | r = 25 + 35
      | c = 
}}
{{eqn | r = 27 + 33
      | c = 
}}
{{eqn | l = 62
      | r = 27 + 35
      | c = 
}}
{{eqn | l = 64
      | r = 9 + 55
      | c = 
}}
{{eqn | r = 15 + 49
      | c = 
}}
{{eqn | r = 25 + 39
      | c = 
}}
{{eqn | l = 66
      | r = 9 + 57
      | c = 
}}
{{eqn | r = 15 + 51
      | c = 
}}
{{eqn | r = 21 + 45
      | c = 
}}
{{eqn | r = 27 + 39
      | c = 
}}
{{eqn | r = 33 + 33
      | c = 
}}
{{eqn | l = 68
      | r = 33 + 35
      | c = 
}}
{{end-eqn}}
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.
{{qed}}