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 A164333 Primes prime(k) such that all integers in the interval [(prime(k-1)+1)/2, (prime(k)-1)/2] are composite numbers. 15
 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, 607, 613, 619, 643 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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 A166308, 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] LINKS V. Shevelev, On critical small intervals containing primes, arXiv:0908.2319 [math.NT], 2009. V. Shevelev, Ramanujan and Labos Primes, Their Generalizations, and Classifications of Primes, J. Int. Seq. 15 (2012) # 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. Integer Seq. 14 (2011) Article 11.6.2. FORMULA {A080359} union {A164294} = {this sequence} union {2,3}. - Vladimir Shevelev, Oct 29 2011 A164368(2) 3 then                 for i from (prevprime(n)+1)/2 to (n-1)/2 do                         if isprime(i) then                                 return false;                         end if;                 end do;                 return true;         else                 false;         end if; end proc: for i from 2 to 700 do         if isA164333(i) then                 printf("%d, ", i);         end if; end do: # R. J. Mathar, Oct 29 2011 MATHEMATICA kmax = 200; Select[Table[{(Prime[k - 1] + 1)/2, (Prime[k] - 1)/2}, {k, 3, kmax}], AllTrue[Range[#[], #[]], CompositeQ]&][[All, 2]]*2 + 1 (* Jean-François Alcover, Nov 14 2017 *) CROSSREFS Cf. A080359, A104272, A164288, A164294, A164332, A001262, A001567, A062568, A141232. Cf. A164368, A164554, A166251, A166252. [Vladimir Shevelev, Oct 10 2009] Sequence in context: A085413 A342943 A244311 * A182365 A069324 A040047 Adjacent sequences:  A164330 A164331 A164332 * A164334 A164335 A164336 KEYWORD nonn AUTHOR Vladimir Shevelev, Aug 13 2009 EXTENSIONS Definition rephrased by R. J. Mathar, Oct 02 2009 STATUS approved

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Last modified July 24 06:56 EDT 2021. Contains 346273 sequences. (Running on oeis4.)