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A166251 Isolated primes: Primes p such that there is no other prime in the interval [2*prevprime(p/2), 2*nextprime(p/2)]. 20
5, 7, 23, 37, 79, 83, 89, 163, 211, 223, 257, 277, 317, 331, 337, 359, 383, 389, 397, 449, 457, 467, 479, 541, 547, 557, 563, 631, 673, 701, 709, 761, 787, 797, 839, 863, 877, 887, 919, 929, 977, 1129, 1181, 1201, 1213, 1237, 1283, 1307, 1327, 1361, 1399, 1409 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Other formulation:  Suppose a prime p >= 5 lies in the interval (2p_k, 2p_(k+1)), where p_n is the n-th prime; p is called isolated if the interval (2p_k, 2p_(k+1)) does not contain any other primes.

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 interval(2p_k, 2p_(k+1)), then 1)if in this interval there are primes only more than p, then p is called a right prime; 2) if in this interval there are primes only less than p, then p is called a left prime; 3) if in this interval there are prime more 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) more than 2 are either right or central primes; the left primes form sequence {13, 19, 31, 43, 53, 61, 73, 103, ...} and all Labos primes (A080359) greater than 3 are either left or central primes.

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

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, J. Integer Seq. 14 (2011) Article 11.6.2

EXAMPLE

Since 2*17 < 37 < 2*19, and the interval (34, 38) does not contain other primes, 37 is an isolated prime.

MATHEMATICA

isolatedQ[p_] := p == NextPrime[2*NextPrime[p/2, -1]] && p == NextPrime[2*NextPrime[p/2], -1]; Select[Prime /@ Range[300], isolatedQ] (* Jean-Fran├žois Alcover, Nov 29 2012, after M. F. Hasler *)

PROG

(Haskell)

a166251 n = a166251_list !! (n-1)

a166251_list = concat $ (filter ((== 1) . length)) $

   map (filter ((== 1) . a010051)) $

   zipWith enumFromTo a100484_list (tail a100484_list)

-- Reinhard Zumkeller, Apr 27 2012

(PARI) is_A166251(n)={n==nextprime(2*precprime(n\2)) & n==precprime(2*nextprime(n/2))}  \\ M. F. Hasler, Oct 05 2012

CROSSREFS

Cf. A166307, A166252, A164368, A104272, A080359, A164333, A164288, A164294, A100484, A182426.

Sequence in context: A214520 A156123 A288908 * A167936 A077242 A121182

Adjacent sequences:  A166248 A166249 A166250 * A166252 A166253 A166254

KEYWORD

nonn

AUTHOR

Vladimir Shevelev, Oct 10 2009, Oct 14 2009

EXTENSIONS

Edited by N. J. A. Sloane, Oct 15 2009

More terms from Alois P. Heinz, Apr 26 2012

Given terms double-checked with new PARI code by M. F. Hasler, Oct 05 2012

STATUS

approved

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Last modified December 2 05:23 EST 2020. Contains 338865 sequences. (Running on oeis4.)