|
| |
|
|
A166251
|
|
Isolated primes: Primes p such that there is no other prime in the interval [2*prevprime(p/2), 2*nextprime(p/2)].
|
|
19
|
|
|
|
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: Two first 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 {13, 19, 31, 43, 53, 61, 73, 103,...} and all Labos primes (A080359) greater than 3 are either left or central primes.
|
|
|
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
|
|
|
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] (* 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.
Sequence in context: A018656 A214520 A156123 * A167936 A077242 A121182
Adjacent sequences: A166248 A166249 A166250 * A166252 A166253 A166254
|
|
|
KEYWORD
|
nonn
|
|
|
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
|
| |
|
|
