%I #10 Apr 17 2021 03:58:20
%S 29,41,59,83,89,113,137,167,173,179,227,233,263,269,317,347,353,359,
%T 467,479,503,557,563,593,641,653,719,773,809,887,977,983,1097,1187,
%U 1193,1283,1307,1367,1433,1439,1487,1493,1523,1619,1697,1823,1907,1997,2063,2153
%N Prime numbers p == 2 (mod 3) such that p-1 has exactly one odd prime divisor and p+1 has exactly one prime divisor > 3 (counting prime divisors with multiplicity in both).
%C Esparza and Gehring (2018) proved that assuming a generalized Hardy-Littlewood conjecture the number of terms not exceeding x is asymptotically (c/2) * x/log(x)^3, where c = A343480 = 5.716497...
%H Amiram Eldar, <a href="/A343479/b343479.txt">Table of n, a(n) for n = 1..10000</a>
%H Carlos Esparza and Lukas Gehring, <a href="https://arxiv.org/abs/1810.08679">Estimating the density of a set of primes with applications to group theory</a>, arXiv:1810.08679 [math.NT], 2018.
%e 29 is a term since it is prime, 29 = 3*9 + 2, 29-1 = 28 = 2^2 * 7 has only one odd prime divisor (7) and 29+1 = 30 = 2*3*5 has only one prime divisor (5) larger than 3.
%t q[n_] := Mod[n, 3] == 2 && PrimeQ[n] && PrimeQ[(n + 1)/2^IntegerExponent[n + 1, 2]/3^IntegerExponent[n + 1, 3]] && PrimeQ[(n - 1)/2^IntegerExponent[n - 1, 2]]; Select[Range[2000], q]
%Y Subsequence of A343478.
%Y Cf. A003627, A038550, A215504, A343480.
%K nonn
%O 1,1
%A _Amiram Eldar_, Apr 16 2021