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%I #48 Jun 01 2023 22:32:28
%S 5,7,23,37,79,83,89,163,211,223,257,277,317,331,337,359,383,389,397,
%T 449,457,467,479,541,547,557,563,631,673,701,709,761,787,797,839,863,
%U 877,887,919,929,977,1129,1181,1201,1213,1237,1283,1307,1327,1361,1399,1409
%N Isolated primes: Primes p such that there is no other prime in the interval [2*prevprime(p/2), 2*nextprime(p/2)].
%C 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.
%C 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.
%C From _Peter Munn_, Jun 01 2023: (Start)
%C The isolated primes are prime(k) such that k-1 and k occur as consecutive terms in A020900.
%C In the tree of primes described in A290183, the isolated primes label the nodes with no sibling nodes.
%C 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).
%C (End)
%H Reinhard Zumkeller, <a href="/A166251/b166251.txt">Table of n, a(n) for n = 1..10000</a>
%H V. Shevelev, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL15/Shevelev/shevelev19.html">Ramanujan and Labos primes, their generalizations, and classifications of primes</a>, J. Integer Seq. 15 (2012) Article 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>, J. Integer Seq. 14 (2011) Article 11.6.2
%e Since 2*17 < 37 < 2*19, and the interval (34, 38) does not contain other primes, 37 is an isolated prime.
%t isolatedQ[p_] := p == NextPrime[2*NextPrime[p/2, -1]] && p == NextPrime[2*NextPrime[p/2], -1]; Select[Prime /@ Range[300], isolatedQ] (* _Jean-François Alcover_, Nov 29 2012, after _M. F. Hasler_ *)
%o (Haskell)
%o a166251 n = a166251_list !! (n-1)
%o a166251_list = concat $ (filter ((== 1) . length)) $
%o map (filter ((== 1) . a010051)) $
%o zipWith enumFromTo a100484_list (tail a100484_list)
%o -- _Reinhard Zumkeller_, Apr 27 2012
%o (PARI) is_A166251(n)={n==nextprime(2*precprime(n\2)) & n==precprime(2*nextprime(n/2))} \\ _M. F. Hasler_, Oct 05 2012
%Y Cf. A166307, A166252, A164368, A104272, A080359, A164333, A164288, A164294, A100484, A182426, A182365.
%Y Cf. A020900, A102820, A290183.
%K nonn,easy
%O 1,1
%A _Vladimir Shevelev_, Oct 10 2009, Oct 14 2009
%E Edited by _N. J. A. Sloane_, Oct 15 2009
%E More terms from _Alois P. Heinz_, Apr 26 2012
%E Given terms double-checked with new PARI code by _M. F. Hasler_, Oct 05 2012
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