%I
%S 29,41,59,83,89,101,113,137,149,167,173,179,197,227,233,251,263,269,
%T 293,317,347,353,359,401,449,467,479,503,557,563,587,593,641,653,677,
%U 719,773,809,887,977,983,1097,1187,1193,1283,1307,1367,1373,1433,1439,1487,1493
%N Prime numbers p == 2 (mod 3) such that p1 has exactly one distinct odd prime divisor and p+1 has exactly one distinct prime divisor > 3.
%C Esparza and Gehring (2018) proved that assuming a generalized HardyLittlewood 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="/A343478/b343478.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, 291 = 28 = 2^2 * 7 has only one distinct odd prime divisor (7) and 29+1 = 30 = 2*3*5 has only one distinct prime divisor (5) larger than 3.
%e 101 is a term since it is prime, 101 = 3*33 + 2, 1011 = 100 = 2^2 * 5^2 has only one distinct odd prime divisor (5) and 101+1 = 102 = 2^2*3*17 has only one distinct prime divisor (17) larger than 3.
%t q[n_] := Mod[n, 3] == 2 && PrimeQ[n] && PrimeNu[(n + 1)/2^IntegerExponent[n + 1, 2]/3^IntegerExponent[n + 1, 3]] == 1 && PrimeNu[(n  1)/2^IntegerExponent[n  1, 2]] == 1; Select[Range[1500], q]
%Y A343479 is a subsequence.
%Y Cf. A003627, A215504, A336101, A343480.
%K nonn
%O 1,1
%A _Amiram Eldar_, Apr 16 2021
