The OEIS is supported by the many generous donors to the OEIS Foundation. Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 60th year, we have over 367,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 A182365 and all Labos primes (A080359) greater than 3 are either left or central primes. From Peter Munn, Jun 01 2023: (Start) The isolated primes are prime(k) such that k-1 and k occur as consecutive terms in A020900. In the tree of primes described in A290183, the isolated primes label the nodes with no sibling nodes. Conjecture: a(n)/A000040(n) is asymptotic to 9. This would follow from my conjectured asymptotic proportion of 1's in A102820 (the first differences of A020900). (End) 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, isolatedQ] (* Jean-François 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, A182365. Cf. A020900, A102820, A290183. Sequence in context: A214520 A156123 A288908 * A167936 A077242 A121182 Adjacent sequences: A166248 A166249 A166250 * A166252 A166253 A166254 KEYWORD nonn,easy 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 11 01:45 EST 2023. Contains 367717 sequences. (Running on oeis4.)