A194598 Union of A080359 and A164294.

2, 3, 13, 19, 31, 43, 53, 61, 71, 73, 101, 103, 109, 113, 131, 139, 151, 157, 173, 181, 191, 193, 199, 229, 233, 239, 241, 251, 269, 271, 283, 293, 311, 313, 349, 353, 373, 379, 409, 419, 421, 433, 439, 443, 463, 491, 499, 509, 523, 571, 577, 593, 599, 601

Offset: 1

Author

Vladimir Shevelev, Aug 30 2011

Keywords

• nonn

Comments

Every greater of twin primes (A006512), beginning with 13, is in the sequence.

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.

This sequence and A164368 are the mutually wrapping up sequences:

a(1) <= A164368(1) <= a(2) <= A164368(2) <= ...

Following the steps to generate T(n,1) in A229608 provides an alternate method of generating this sequence. - Bob Selcoe, Oct 27 2015

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• V. Shevelev, Ramanujan and Labos primes, their generalizations and classifications of primes, arXiv:0909.0715 [math.NT], 2009, 2011.
• V. Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq. 15 (2012) Article 12.5.4
• J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, arXiv:1105.2249 [math.NT], 2011.
• J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, J. Integer Seq. 14 (2011) Article 11.6.2.

Crossrefs

Cf. A080359, A164294, A164368, A193507, A194186, A212493, A212541, A229608.

If the first two terms are omitted we get A164333.

Programs

Mathematica

primePiMax = 200;
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 *)

Formula

First column of array A229608. - Bob Selcoe, Oct 27 2015

For n >= 3, a(n) = A164333(n-2). - Peter Munn, Aug 30 2017