Even Integers not Expressible as Sum of 3, 5 or 7 with Prime

Theorem
The even integers that cannot be expressed as the sum of $2$ prime numbers where one of those primes is $3$, $5$ or $7$ begins:
 * $98, 122, 124, 126, 128, 148, 150, \ldots$

Proof
These are the primes which coincide with the upper end of a prime gap greater than $6$.

These can be found at:
 * $89$ to $97$: prime gap of $8$
 * $113$ to $127$: prime gap of $14$
 * $139$ to $149$: prime gap of $10$

and so on.

We have that:

and so on.