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%I #18 Aug 21 2019 06:02:05
%S 3,5,11,23,179,239,359,719,5039,55439,665279,6486479,32432399,
%T 698377679,735134399,1102701599,20951330399,3212537327999,
%U 149602080797769599,299204161595539199,2718551763981393634806325317503999
%N Primes p such that 2*p + 1 is also prime and p + 1 is a highly composite number (definition 1).
%C An equivalent definition of this sequence: odd Sophie Germain primes that differ from a highly composite number by 1.
%C Intersection of A005384 (Sophie Germain primes) and A072828.
%C With the exception of 5, a subsequence of A002515 (Lucasian primes).
%C Except for first two terms, this is a subsequence of A054723.
%C Except for n = 2, 2*a(n) + 1 is a prime factor of A000225(a(n)) (i.e., 2*23 + 1 divides 2^23 - 1).
%C Conjecture: for n >= 5, GCD(A000032(a(n)), A000225(a(n))) = 2*a(n) + 1.
%H Amiram Eldar, <a href="/A214873/b214873.txt">Table of n, a(n) for n = 1..25</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Sophie_Germain_prime">Sophie Germain prime</a>
%e 23 is a term because both 23 and 47 are primes and also 24 is a highly composite number.
%t lst = {}; a = 0; Do[b = DivisorSigma[0, n + 1]; If[b > a, a = b; If[PrimeQ[n] && PrimeQ[2*n + 1], AppendTo[lst, n]]], {n, 1, 10^6, 2}]; lst
%Y Cf. A054723.
%K nonn
%O 1,1
%A _Arkadiusz Wesolowski_, Jul 30 2012