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%I #30 Jun 01 2023 22:32:32
%S 13,19,31,43,53,61,71,73,101,103,109,113,131,139,151,157,173,181,191,
%T 193,199,229,233,239,241,251,269,271,283,293,311,313,349,353,373,379,
%U 409,419,421,433,439,443,463,491,499,509,523,571,577,593,599,601,607,613,619,643
%N Primes prime(k) such that all integers in the interval [(prime(k-1)+1)/2, (prime(k)-1)/2] are composite numbers.
%C Let p_k be the k-th prime. A prime p is in the sequence iff the interval of the form (2p_k, 2p_(k+1)), containing p, also contains a prime less than p. The sequence is connected with the following classification of primes: the first two primes 2,3 form a separate set of primes; let p >= 5 be in the interval (2p_k, 2p_(k+1)), then 1) if in this interval there are only primes greater than p, then p is called a right prime; 2) if in this interval there are only primes less than p, then p is called a left prime; 3) if in this interval there are primes both greater and less than p, then p is called a central prime; 4) if this interval does not contain other primes, then p is called an isolated prime. In particular, the right primes form sequence A166307, and all Ramanujan primes (A104272) greater than 2 are either right or central primes; the left primes form sequence A182365, and all Labos primes (A080359) greater than 3 are either left or central primes; the central primes form A166252 and the isolated primes form A166251. [_Vladimir Shevelev_, Oct 10 2009] [Sequence reference updated by _Peter Munn_, Jun 01 2023]
%C Disjoint union of A166252 and A182365. - _Peter Munn_, Jun 01 2023 [an edited version of a contribution by _Vladimir Shevelev_ in 2009]
%H V. Shevelev, <a href="http://arXiv.org/abs/0908.2319">On critical small intervals containing primes</a>, arXiv:0908.2319 [math.NT], 2009.
%H V. Shevelev, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Shevelev/shevelev19.html">Ramanujan and Labos Primes, Their Generalizations, and Classifications of Primes</a>, J. Int. Seq. 15 (2012) # 12.5.4
%H J. Sondow, J. W. Nicholson, and T. D. Noe, <a href="http://arxiv.org/abs/1105.2249"> Ramanujan Primes: Bounds, Runs, Twins, and Gaps</a>, arXiv:1105.2249 [math.NT], 2011; J. Integer Seq. 14 (2011) Article 11.6.2.
%F {A080359} union {A164294} = {this sequence} union {2,3}. - _Vladimir Shevelev_, Oct 29 2011
%F A164368(2)<a(1) < A164368(3)<a(2) < A164368(4)<a(3)<... [_Vladimir Shevelev_, Oct 10 2009]
%e Let p=53. We see that 2*23<53<2*29. Since the interval (46, 58) contains prime 47<53 and does not contain any prime more than 53, then, by the considered classification 53 is left prime and it is in the sequence. [_Vladimir Shevelev_, Oct 10 2009]
%p isA164333 := proc(n)
%p local i ;
%p if isprime(n) and n > 3 then
%p for i from (prevprime(n)+1)/2 to (n-1)/2 do
%p if isprime(i) then
%p return false;
%p end if;
%p end do;
%p return true;
%p else
%p false;
%p end if;
%p end proc:
%p for i from 2 to 700 do
%p if isA164333(i) then
%p printf("%d,",i);
%p end if;
%p end do: # _R. J. Mathar_, Oct 29 2011
%t kmax = 200; Select[Table[{(Prime[k - 1] + 1)/2, (Prime[k] - 1)/2}, {k, 3, kmax}], AllTrue[Range[#[[1]], #[[2]]], CompositeQ]&][[All, 2]]*2 + 1 (* _Jean-François Alcover_, Nov 14 2017 *)
%Y Cf. A080359, A104272, A164288, A164294, A164332, A001262, A001567, A062568, A141232.
%Y Cf. A164368, A164554, A166251, A166252. [_Vladimir Shevelev_, Oct 10 2009]
%Y Cf. A166307, A182365.
%K nonn
%O 1,1
%A _Vladimir Shevelev_, Aug 13 2009
%E Definition rephrased by _R. J. Mathar_, Oct 02 2009
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