A215238 -id:A215238 - cp's OEIS frontend

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.

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

A104380 Smallest prime p(i) such that between 2p(i) and 2p(i+1) there exist n primes.

Original entry on oeis.org

59, 2, 5, 31, 89, 509, 113, 1129, 1951, 7253, 19609, 16141, 13339, 28229, 1327, 281431, 396733, 1122287, 461717, 370261, 2010733, 492113, 3279841, 14246971, 3117299, 5826001, 20831323, 47326693, 60487759, 189695659, 191912783, 1301171033, 2044207843, 3129752191, 476956933, 836806669, 2433630109

Offset: 0

Ali A. Tanara (tanara(AT)khayam.ut.ac.ir) and Robert G. Wilson v, Mar 03 2005

nonn

			a(0)=59 because between 2*59=118 and 2*61=122 there are no primes and 59 is the least prime with this characteristic.
a(4)=89 because between 2*89=178 and 2*97=194 there are 4 primes, namely 179, 181, 191 and 193.

Charles R Greathouse IV, Table of n, a(n) for n = 0..53

Cf. A060756, A102820, A215238.

f[n_] := PrimePi[2Prime[n + 1]] - PrimePi[2Prime[n]]; t = Table[0, {30}]; Do[a = f[n]; If[t[[a + 1]] == 0, t[[a + 1]] = Prime[n]], {n, 2500000}]; t

ct(a,b)=sum(k=a,b,isprime(k))
a(n)=my(p=2); forprime(q=3,, if(ct(2*p+1,2*q-1)==n, return(p)); p=q) \\ Charles R Greathouse IV, Nov 05 2017

a(27)-a(36) from Charles R Greathouse IV, Nov 05 2017

A215237 Least number k for which primepi(prime(k+1)/2) - primepi(prime(k)/2) = n.

Original entry on oeis.org

1, 2, 30, 259, 429, 4612, 26466, 88110, 31545, 104071, 2775456, 14614604, 15793779, 164082567, 476853784, 495207013, 3613011290, 9032608100, 69827848342

Offset: 0

T. D. Noe, Oct 09 2012

See A215238 and A215239 for prime(a(n)) and the next prime.

Equivalently stated, a(n) is least k such that there are exactly n primes between prime(k)/2 and prime(k+1)/2. - Peter Munn, May 20 2019

			For n = 2, the consecutive primes are 113 and 127; there are two primes between 56.5 and 63.5.  For n = 3, the consecutive primes are 1637 and 1657; there are three primes between 818.5 and 828.5.

Cf. A217564, A215238, A215239.

t = Table[PrimePi[Prime[n+1]/2] - PrimePi[Prime[n]/2], {n, 100000}]; Flatten[Table[Position[t, n, 1, 1], {n, 0, 8}]]

a(14)-a(18) from Donovan Johnson, Oct 13 2012

A215239 Prime number following prime(A215237).

Original entry on oeis.org

3, 5, 127, 1657, 2999, 44351, 305717, 1133131, 370373, 1357333, 46006967, 268119667, 291057563, 3429782399, 10502593369, 10926444847, 87241771051, 226751019863, 1901687257829

Offset: 0

T. D. Noe, Oct 11 2012

We use offset 0 because A215237 uses that offset.

Cf. A215237, A215238.

t = Table[PrimePi[Prime[n + 1]/2] - PrimePi[Prime[n]/2], {n, 100000}]; t2 = Flatten[Table[Position[t, n, 1, 1], {n, 0, 8}]]; NextPrime[Prime[t2]]

a(14)-a(18) from Donovan Johnson, Oct 13 2012

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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.

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A104380 Smallest prime p(i) such that between 2p(i) and 2p(i+1) there exist n primes.

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A215237 Least number k for which primepi(prime(k+1)/2) - primepi(prime(k)/2) = n.

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A215239 Prime number following prime(A215237).

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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.