Primes p such that ((3/2)*(p-1)+1) and (3*p-2) are prime.
5, 13, 53, 181, 293, 401, 461, 541, 673, 853, 953, 1021, 1453, 1481, 1693, 1861, 1901, 2113, 2213, 2221, 2333, 2473, 2741, 3061, 3221, 3301, 3613, 3833, 4153, 4201, 4241, 4253, 4561, 4813, 5081, 5641, 5693, 6073, 6101, 6133, 6481, 7001, 7193, 7541, 7853
1
These p appear the produce the highest points in the plot of S000733. That is, it appears that for these primes p, p*((3/2)(p-1)+1)*(3*p-2) is the GCD of the numbers binomial(3*p, 3*k) for k = 1..p-1.
T. D. Noe, Plot of 1000 terms
T. D. Noe, Table of 1000 terms
(Mma) t = {}; p = 1; While[Length[t] < 50, p = NextPrime[p]; If[PrimeQ[(3/2)*(p-1)+1] && PrimeQ[3*p-2], AppendTo[t, p]]]; t
Cf. S000733.
nonn
T. D. Noe, Oct 27 2015