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For number 152, 152 = 3+149 = 13+139 = 43+109 = 73+79. The differences of adjacent smaller primes in each of the decomposition groups are 10, 30, 30 respectively. None of them is 6 or 12. So 152 is included.
n = 0; Table[
While[n++; s = 2*n; ct = 0; p = 1;
While[p = NextPrime[p]; p <= n,
If[PrimeQ[s - p], ok = 0; a1 = p - 12; b1 = s - a1; a2 = p - 6;
b2 = s - a2; a3 = p + 6; b3 = s - a3; a4 = p + 12; b4 = s - a4;
If[a1 > 0, If[PrimeQ[a1] && PrimeQ[b1], ok = 1]];
If[a2 > 0, If[PrimeQ[a2] && PrimeQ[b2], ok = 1]];
If[a3 <= n, If[PrimeQ[a3] && PrimeQ[b3], ok = 1]];
If[a4 <= n, If[PrimeQ[a4] && PrimeQ[b4], ok = 1]];