Largest Even Integer not expressible as Sum of 2 k Odd Composite Integers

Theorem
Let $k \in \Z_{>0}$ be a (strictly) positive integer.

The largest even integer which cannot be expressed as the sum of $2 k$ odd positive composite integers is $18 k + 20$.

Proof
Let $n$ be an even integer greater than $18 k + 20$.

Then $n - 9 \paren {2 k - 2}$ is an even integer greater than $18 k + 20 - 9 \paren {2 k - 2} = 38$.

By Positive Even Integers not Expressible as Sum of 2 Composite Odd Numbers, every even integer greater than $38$ can be expressed as the sum of $2$ odd positive composite integers.

Thus $n - 9 \paren {2 k - 2}$ can be expressed as the sum of $2$ odd positive composite integers.

So, let $a$ and $b$ be odd composite integers such that $a + b = n - 9 \paren {2 k - 2}$.

Then:
 * $9 \paren {2 k - 2} + a + b = n$

This is an expression for $n$ as the sum of $2 k$ odd positive composite integers, in which $2 k - 2$ of them are occurrences of $9$.

Thus such an expression can always be found for $n > 18 k + 20$.

It remains to be shown that $18 k + 20$ is not expressible as the sum of $2 k$ odd positive composite integers.

The $k = 1$ case is demonstrated in Positive Even Integers not Expressible as Sum of 2 Composite Odd Numbers.

Now suppose $k \ge 2$.

The two smallest odd positive composite integers are $9$ and $15$.

Suppose $18 k + 20$ is expressible as the sum of $2 k$ odd positive composite integers.

Then at least $2 k - 3$ of them are $9$'s, because:


 * $9 \paren {2 k - 4} + 4 \times 15 = 18 k + 24 > 18 k + 20$

Then the problem reduces to finding an expression of $18 k + 20 - 9 \paren {2 k - 3} = 47$ as the sum of $3$ odd positive composite integers.

The first few odd positive composite integers are:


 * $9, 15, 21, 25, 27, 33, 35, 39, 45$

Their difference with $47$ are:


 * $38, 32, 26, 22, 20, 14, 12, 8, 2$

The integers above are in the set of integers not expressible as a sum of $2$ odd positive composite integers.

The full set of these can be found in Positive Even Integers not Expressible as Sum of 2 Composite Odd Numbers.

Thus $18 k + 20$ is not expressible as the sum of $2 k$ odd positive composite integers.

This proves the result.