A290183 -id:A290183 - cp's OEIS frontend

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

Vladimir Shevelev, Oct 10 2009, Oct 14 2009

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)

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

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

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

Cf. A020900, A102820, A290183.

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

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 *)

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

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

A080192 Complement of A080191 relative to A000040. Prime p is a term iff there is no prime between 2p and 2q, where q is the next prime after p.

Original entry on oeis.org

59, 71, 101, 107, 149, 263, 311, 347, 461, 499, 521, 569, 673, 757, 821, 823, 857, 881, 883, 907, 967, 977, 1009, 1061, 1091, 1093, 1151, 1213, 1279, 1283, 1297, 1301, 1319, 1433, 1487, 1489, 1493, 1549, 1571, 1597, 1619, 1667, 1697, 1721, 1787, 1871, 1873

Offset: 1

Klaus Brockhaus, Feb 10 2003

nonn

From Peter Munn, Oct 19 2017: (Start)
This is also a list of the leaf node labels in the tree of primes described in A290183.
For k > 0, the earliest run of k adjacent primes in this sequence starts with the least prime greater than A215238(k+1)/2. Thus we see that A215238(3) = 1637 corresponds to 821 followed by 823 being the first run of 2 adjacent primes in this sequence.
(End)
From Peter Munn, Nov 02 2017: (Start)
If p is in A005384 (a Sophie Germain prime), 2p+1 is therefore a prime, so p cannot be in this sequence. Similarly, any prime p in A023204 has a corresponding prime 2p+3, which (if p>2) likewise implies its absence (and if p=2 it is in A005384).
If p is the lesser of twin primes it is in this sequence if it is neither Sophie Germain nor in A023204.
Conjecture: a(n)/A000040(n) is asymptotic to 3. Reason: I expect the distribution of terms in A102820 to converge to a geometric distribution with mean value 2.
(End)

			59 is a term since 113 is the prime preceding 2*59, 127 is the next prime and 61 is the largest of all prime factors of 114, ..., 122 = 2*61, ..., 126.

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Michel Marcus)

A080191 is the complement of this sequence relative to A000040.
Sequences with related analysis: A005384, A023204, A052248, A102820, A215238, A290183.
Sequences with similar definitions: A195270, A195271, A195325, A195377.

Select[Prime[Range[300]],NextPrime[2#]>2NextPrime[#]&] (* Harvey P. Dale, Jul 07 2011 *)

¯1↓b/⍨(1⌽a)<1πa←2×b←¯2π⍳1E4 ⍝ Michael Turniansky, Dec 29 2020

{forprime(k=2,1873,p=precprime(2*k); q=nextprime(p+1); m=0; for(j=p+1,q-1,f=factor(j); a=f[matsize(f)[1],1]; if(m

isok(p) = isprime(p) && (primepi(2*p) == primepi(2*nextprime(p+1)));
forprime(p=2, 2000, if (isok(p), print1(p, ", "))) \\ Michel Marcus, Sep 22 2017

first(n) = my(res = vector(n), i = 0); {n==0&&return([]); forprime(p = 2, , if(nextprime(2*p) > 2*nextprime(p + 1), i++; res[i] = p; if(i == n, return(res))))} \\ David A. Corneth, Oct 25 2017

For all k, prime(k) = A000040(k) is a term if and only if A102820(k) = 0. - Peter Munn, Oct 24 2017

A102820 Number of primes between 2prime(n) and 2prime(n+1), where prime(n) is the n-th prime.

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 1, 2, 2, 2, 3, 1, 1, 1, 3, 3, 0, 2, 2, 0, 3, 1, 2, 4, 2, 0, 1, 0, 1, 6, 1, 3, 1, 3, 0, 3, 1, 1, 1, 3, 1, 3, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 3, 2, 2, 0, 1, 1, 1, 1, 3, 6, 2, 0, 1, 6, 1, 3, 0, 1, 1, 3, 2, 2, 1, 2, 1, 1, 2, 4, 1, 3, 1, 1, 2, 1, 2, 1, 0, 1, 4, 2, 1, 3, 0, 2, 5, 0, 5, 3, 3, 2, 1, 0, 2

Offset: 1

Ali A. Tanara (tanara(AT)khayam.ut.ac.ir), Feb 27 2005

Number of primes between successive even semiprimes. [Juri-Stepan Gerasimov, May 01 2010]
From Peter Munn, Jun 01 2023: (Start)
First differences of A020900.
A080192 lists prime(n) corresponding to the zero terms.
A104380(k) is prime(n) corresponding to the first occurrence of k as a term.
If a(n) is nonzero, A059786(n) is the smallest and A059788(n+1) the largest of the a(n) enumerated primes. In the tree of primes described in A290183, these primes label the child nodes of prime(n).
Conjecture: the asymptotic proportions of 0's, 1's, ... , k's, ... are 1/3, 2/9, ... , 2^k/3^(k+1), ... .
(End)

			a(15)=3 because there are 3 primes between the doubles of the 15th and 16th primes, that is between 2*47 and 2*53.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
V. Shevelev, On critical small intervals containing primes, arXiv:0908.2319 [math.NT], 2009.

Sequences with related analysis: A020900, A059786, A059788, A080192, A104380, A290183.
Cf. A104272, A080359. [Vladimir Shevelev, Aug 24 2009]
Cf. A100484, A010051.
Sequences with similar definitions: A104289, A217564.

a102820 n = a102820_list !! (n-1)
a102820_list =  map (sum . (map a010051)) $
   zipWith enumFromTo a100484_list (tail a100484_list)
-- Reinhard Zumkeller, Apr 29 2012

Table[PrimePi[2 Prime[n+1]]-PrimePi[2 Prime[n]], {n, 150}] (* Zak Seidov *)
Differences[PrimePi[2 Prime[Range[110]]]] (* Harvey P. Dale, Oct 29 2022 *)

a(n) = primepi(2*prime(n+1)) - primepi(2*prime(n)); \\ Michel Marcus, Sep 22 2017

a(n) = A020900(n+1) - A020900(n). - Peter Munn, Jun 01 2023

More terms from Zak Seidov, Feb 28 2005

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Extensions

A080192 Complement of A080191 relative to A000040. Prime p is a term iff there is no prime between 2*p and 2*q, where q is the next prime after p.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

NARS2000

PARI

PARI

PARI

Formula

A102820 Number of primes between 2*prime(n) and 2*prime(n+1), where prime(n) is the n-th prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions

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

A080192 Complement of A080191 relative to A000040. Prime p is a term iff there is no prime between 2p and 2q, where q is the next prime after p.

A102820 Number of primes between 2prime(n) and 2prime(n+1), where prime(n) is the n-th prime.