This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A167917 #9 Feb 06 2021 08:31:20
%S A167917 3,7,31,8191,524287
%N A167917 Mersenne primes that belong to Cunningham chains = {3, 7} U {Mp | 2Mp - 1 is prime. (Mp a Mersenne prime)}.
%C A167917 If p is prime, p >= 5, and Mp belongs to a chain, Mp is always the first term of a chain of the second kind. This is true since (Mp+1)/2 = (2^p - 1 +1)/2 = 2^(p-1), which is composite for p >= 3. (Mp-1)/2 = (2^p - 1 -1)/2 = 2^(p-1)-1 = a. For p >= 5, a is composite since a>3, and a mod 3 = 0. Finally 2Mp + 1 = 2(2^p - 1)+1 = 2^(p+1)-1 = a. If p>=3, a is composite because a > 3, and a mod 3 = 0. We can conclude that beginning with 31, a Mersenne prime can only starts a Cunningham chain of the second kind. If Mp >= 31 starts a chain, the second term of this chain is 2Mp -1=2(2^p - 1)-1 = 2^(p+1) - 3.
%C A167917 That is a number of the form 2^N - 3, even N, so also of the form a^2 - 3, a = 2^(N/2). In this case any factor f of the second term of a chain satisfies f mod 24=1, or f mod 24=11, or f mod 24=13, or f mod 24=23. (1) The next term of this sequence is an unknown Mersenne prime. Probably many primes of this kind will be determined until this term be found. In the work with the known Mersenne primes, M42643801 gives T=2^(42643801+1) -3. The smallest factor of T is f = 38334482051, which is greater than 2^35.
%C A167917 Considering the probabilities given in the second reference, one can conclude that before T was identified as composed (by the exam of all the primes less than f satisfying (1)), the probability of prime T reached a value of 1 in 609,197. This probability is small, but not negligible. Note that the largest known Cunningham chain of length 2 has starting prime 607095* 2^176311 - 1. This is a "very small chain" compared with a chain beginning with a new Mersenne prime.
%H A167917 Niccolò Castronuovo, <a href="https://arxiv.org/abs/2102.02739">On the number of fixed points of the map gamma</a>, arXiv:2102.02739 [math.NT], 2021. Mentions this sequence.
%H A167917 Mathpages, <a href="http://www.mathpages.com/home/kmath480/kmath480.htm">Some Properties of the Lucas Sequence</a>
%H A167917 GIMPS, <a href="http://www.mersenne.org/various/math.php">Mathematics and Research Strategy</a>
%H A167917 Wikipedia, <a href="http://en.wikipedia.org/wiki/Cunningham_chain">Cunningham chain</a>
%e A167917 a(1) = 3 since 2*3 - 1 = 5. a(2) = 7 because 2*7 - 1 = 13.
%Y A167917 Cf. A000043, A000668, A050415, A050414.
%K A167917 nonn,hard,more
%O A167917 1,1
%A A167917 _Washington Bomfim_, Nov 15 2009