1,1

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)

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

Vladimir Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq. 15 (2012) Article 12.5.4

Jonathan 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

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

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 *)

(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

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

nonn,easy

Vladimir Shevelev, Oct 10 2009, Oct 14 2009

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

approved