Conjecture on OEIS A167055

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OEIS A167055 Numbers n such that $12n + 5$ is prime.

$0, 1, 2, 3, 4, 7, 8, 9, 11, 12, 14, 16, 19, 21,...$ are items of OEIS A167055. I conjecture that the set of the sum of every two items of this sequence is the set of nonnegative integers. i.e.:$0+0=0,\ 0+1=1,\ ...,\ 1+4=5,...$

Further information for A167055 not in OEIS A167055: nonnegative integers that not of the following two forms: $$ \begin{align*} &3x^2+(6y−3)x−y\\ &3x^2+(6y−3)x+(y−1) ,\ x,y \in \mathbb{Z^+} \end{align*} $$

Is there some clue to solve this problem?

More information see: How come if i not of the following form, then 12i+5 must be prime?