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A194598 Union of A080359 and A164294.

%I A194598 #51 Aug 18 2018 08:37:47
%S A194598 2,3,13,19,31,43,53,61,71,73,101,103,109,113,131,139,151,157,173,181,
%T A194598 191,193,199,229,233,239,241,251,269,271,283,293,311,313,349,353,373,
%U A194598 379,409,419,421,433,439,443,463,491,499,509,523,571,577,593,599,601
%N A194598 Union of A080359 and A164294.
%C A194598 Every greater of twin primes (A006512), beginning with 13, is in the sequence.
%C A194598 A very simple sieve for the generation of the terms is the following: Let p_n be the n-th prime. Consider consecutive intervals of the form (2p_n, 2p_{n+1}), n=1,2,... From every interval containing at least one prime we take the first one and remove it from the set of all primes. Then all remaining primes form the sequence. Let us demonstrate this sieve: For primes 2,3,5,7,11,... consider intervals (4,6), (6,10), (10,14), (14,22), (22,26), (26,34), ... . Removing from the set of all primes the first prime of each interval, i.e., 5,7,11,17,23,29,... ,we obtain 2,3,13,19,31, etc.
%C A194598 This sequence and A164368 are the mutually wrapping up sequences:
%C A194598    a(1) <= A164368(1) <= a(2) <= A164368(2) <= ...
%C A194598 Following the steps to generate T(n,1) in A229608 provides an alternate method of generating this sequence. - _Bob Selcoe_, Oct 27 2015
%H A194598 Alois P. Heinz, <a href="/A194598/b194598.txt">Table of n, a(n) for n = 1..1000</a>
%H A194598 V. Shevelev, <a href="http://arxiv.org/abs/0909.0715">Ramanujan and Labos primes, their generalizations and classifications of primes</a>, arXiv:0909.0715 [math.NT], 2009, 2011.
%H A194598 V. Shevelev, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL15/Shevelev/shevelev19.html">Ramanujan and Labos primes, their generalizations, and classifications of primes</a>, J. Integer Seq. 15 (2012) Article 12.5.4
%H A194598 J. Sondow, J. W. Nicholson, and T. D. Noe, <a href="https://arxiv.org/abs/1105.2249">Ramanujan Primes: Bounds, Runs, Twins, and Gaps</a>, arXiv:1105.2249 [math.NT], 2011.
%H A194598 J. Sondow, J. W. Nicholson, and T. D. Noe, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Noe/noe12.html">Ramanujan Primes: Bounds, Runs, Twins, and Gaps</a>, J. Integer Seq. 14 (2011) Article 11.6.2.
%F A194598 First column of array A229608. -  _Bob Selcoe_, Oct 27 2015
%F A194598 For n >= 3, a(n) = A164333(n-2). - _Peter Munn_, Aug 30 2017
%t A194598 primePiMax = 200;
%t A194598 Join[{2, 3}, Select[Table[{(Prime[k-1] + 1)/2, (Prime[k] - 1)/2}, {k, 3, primePiMax}], AllTrue[Range[#[[1]], #[[2]]], CompositeQ]&][[All, 2]]*2+1] (* _Jean-François Alcover_, Aug 18 2018 *)
%Y A194598 Cf. A080359, A164294, A164368, A193507, A194186, A212493, A212541, A229608.
%Y A194598 If the first two terms are omitted we get A164333.
%K A194598 nonn
%O A194598 1,1
%A A194598 _Vladimir Shevelev_, Aug 30 2011