A166251 Isolated primes: Primes p such that there is no other prime in the interval [2*prevprime(p/2), 2*nextprime(p/2)].

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

Offset: 1

Author

Vladimir Shevelev, Oct 10 2009, Oct 14 2009

Keywords

• nonn
• easy

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)

Examples

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

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

Crossrefs

Cf. A166307, A166252, A164368, A104272, A080359, A164333, A164288, A164294, A100484, A182426, A182365.

Cf. A020900, A102820, A290183.

Programs

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

Mathematica

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 *)

PARI

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

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