

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
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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 nth 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.
The isolated primes are prime(k) such that k1 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



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] (* JeanFrançois Alcover, Nov 29 2012, after M. F. Hasler *)


PROG

(Haskell)
a166251 n = a166251_list !! (n1)
a166251_list = concat $ (filter ((== 1) . length)) $
map (filter ((== 1) . a010051)) $
zipWith enumFromTo a100484_list (tail a100484_list)
(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.


KEYWORD

nonn,easy


AUTHOR



EXTENSIONS

Given terms doublechecked with new PARI code by M. F. Hasler, Oct 05 2012


STATUS

approved



