Imagine that we have a triangle that starts with 2,3 and grows like Pascal's triangle but instead uses the smallest prime $\geq$ to the sum of the above two primes. Visually:
$ \begin{array}{ccccccccc} &&& 2 && 3 \\ && 2 && 5 && 3 \\ & 2 && 7 && 11 && 3 \\ 2 && 11 && 19 && 17 && 3 \end{array} $
Obviously, all primes will be seen in the first diagonal on the left side. However, if we neglect that diagonal, then there are some primes that never appear. The sequence starts: $5,7,13,43,73,103,109,\dots$
One might notice that all of these primes are second members of twin prime pairs. They would be correct. In fact, the diagonal on the right side does capture every prime except for those that are the second prime of a twin prime pair because $p+3$ skips over $p+2$. However...there ARE some hold-outs. There are primes that do not appear in the main body of the triangle, but DO appear in the right side diagonal despite not being a twin prime. In other words, they appear only twice in the entire triangle while having composite odd neighbors. The first few are $23, 53, 79, 83, 131, 157, \dots$.
Is there something special about these primes that explains this curious fact?