What Are the Possible Sums of Prime Number Pairs?

Loren Booda
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List all of the possible sums of prime number pairs with each element taken once.

For instance: 2+3=5, 2+5=7, 3+5=8, 2+7=9, 3+7=10, 5+7=12, 5+11=16, 5+13=18 . . .

Can you find significance in this progression? Have you seen this sequence before?
 
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That "sequence" will likely consist of every even number that is not of the form 2p as well as every odd number of the form p+2 (p is a prime number). Are you familiar with the Goldbach and Levy conjectures?

If you have a more specific question than the somewhat open-ended question in the OP, please get straight to the specific question.
 
I'm sorry, I thought I had found a more fundamental pattern.
 

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