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%I #31 Jun 18 2023 11:29:14
%S 5,41,1031,3299,3761,6323,7649,7841,10589,12959,18443,37853,38039,
%T 44729,58511,59723,60509,77813,105971,134489,137999,190523,194933,
%U 200351,239633,250751,266093,288653,315011,366161,376241,377633,393401,410999,420419,420779,450431,474983
%N Prime-indexed primes p such that 2*p + 1 is also a prime-indexed prime.
%C Prime-indexed Sophie Germain primes whose associated safe primes are also prime-indexed. Subsequence of A005384.
%H Harvey P. Dale, <a href="/A318294/b318294.txt">Table of n, a(n) for n = 1..300</a>
%H N. Fernandez, <a href="http://www.borve.org/primeness/FOP.html">An order of primeness, F(p)</a>
%H N. Fernandez, <a href="/A006450/a006450.html">An order of primeness</a> [cached copy, included with permission of the author]
%e 5 = prime(3) is a prime-indexed prime, and 2*5 + 1 = 11 = prime(5) is also a prime-indexed prime. Therefore 5 is in the sequence.
%e 3 = prime(2) is a prime-indexed prime, but 2*3 + 1 = 7 = prime(4) is not a prime-indexed prime since 4 is composite. Therefore 3 is not in the sequence.
%t Select[Table[Prime[p],{p,Prime[Range[5000]]}],AllTrue[{2#+1,PrimePi[2#+1]},PrimeQ]&] (* _Harvey P. Dale_, Jun 18 2023 *)
%o (PARI) isok(p) = isprime(p) && isprime(primepi(p)) && isprime(2*p+1) && isprime(primepi(2*p+1)); \\ _Michel Marcus_, Aug 28 2018
%Y Cf. A005384, A006450.
%K nonn
%O 1,1
%A _David James Sycamore_, Aug 23 2018
%E More terms from _Michel Marcus_, Aug 28 2018
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