A195377 -id:A195377 - 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

A218769 Let (p,p+2) be the n-th twin prime pair. a(n) is the least integer r > 1 for which the interval (rp, r(p+2)) contains no primes, or a(n)=0, if no such r exists.

Original entry on oeis.org

0, 0, 0, 0, 4, 0, 2, 2, 2, 2, 3, 2, 5, 5, 4, 5, 4, 4, 3, 2, 2, 4, 4, 2, 2, 2, 6, 3, 3, 4, 3, 2, 3, 2, 2, 7, 3, 3, 2, 2, 2, 6, 0, 3, 2, 2, 5, 5, 23, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 5, 2

Offset: 1

Vladimir Shevelev, Peter J. C. Moses, and Charles R Greathouse IV, Nov 05 2012

nonn

For n<=20000, the largest a(n) is a(49)=23. a(n)=0 for n = 1, 2, 3, 4, 6, 43, 37890, 606457, ... corresponding to the twin primes (p, p+2) with p=3, 5, 11, 17, 41, 1277, 5995727, 143556431, ....

			The 13th twin prime pair is {179, 181}. For r = 2 the range {358, ..., 362} contains prime 359; for r = 3, the range {537, ..., 543} contains prime 541; for r = 4, the range {716, ..., 724} contains prime 719. But for r = 5, the range {895, ..., 905} does not contain any prime. Thus a(13) = 5.

Peter J. C. Moses, Table of n, a(n) for n = 1..20000
Vladimir Shevelev, Charles R. Greathouse IV, Peter J. C. Moses, On intervals (kn, (k+1)n) containing a prime for all n>1, Journal of Integer Sequences, Vol. 16 (2013), Article 13.7.3. arXiv:1212.2785
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. A218279, A001359, A218561, A195270, A195271, A195325, A195377, A195379.

rmax = 100; p1[1] = 3; p1[n_] := p1[n] = (p = NextPrime[p1[n-1]]; While[ !PrimeQ[p+2], p = NextPrime[p]]; p); a[n_] := Catch[ For[r = 2, r <= rmax, r++, If[ PrimePi[r*p1[n]] == PrimePi[r*(p1[n] + 2)], Throw[r], If[r == rmax, Throw[0]]]]]; Table[ a[n] , {n, 1, 65}] (* Jean-François Alcover, Dec 13 2012 *)

Typo in definition corrected by Jonathan Sondow, Dec 21 2012

A195379 3.5-gap primes: Primes prime(k) such that there is no prime between 7prime(k)/2 and 7prime(k+1)/2.

Original entry on oeis.org

2, 137, 281, 521, 641, 883, 937, 1087, 1151, 1229, 1277, 1301, 1489, 1567, 1607, 1697, 2027, 2081, 2237, 2381, 2543, 2591, 2657, 2687, 2729, 2801, 2851, 2969, 3119, 3257, 3301, 3359, 3463, 3467, 3529, 3673, 3733, 3793, 3821, 3851, 4073, 4217, 4229, 4241, 4259, 4283, 4337, 4421, 4481

Offset: 1

Vladimir Shevelev, Sep 17 2011

nonn

Cf. A080192, A195270, A195271, A195377, A195325, A195329, A193507, A194186, A164368, A194598, A194658, A194659, A194674, A164288, A164294.

Select[Prime[Range[1000]], PrimePi[7*NextPrime[#]/2] == PrimePi[7*#/2] &] (* T. D. Noe, Sep 20 2011 *)

Corrected by R. J. Mathar, Sep 20 2011

A218561 4-gap primes: Prime p is a term iff there is no prime between 4p and 4nextprime(p), where nextprime=A151800.

Original entry on oeis.org

29, 71, 137, 197, 239, 269, 347, 419, 431, 641, 659, 809, 821, 1061, 1091, 1151, 1289, 1489, 1607, 1721, 1783, 1877, 1949, 1993, 2083, 2141, 2267, 2339, 2381, 2389, 2549, 2729, 2801, 2833, 2969, 2999, 3019, 3041, 3217, 3253, 3299, 3329, 3389, 3461

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Nov 02 2012

nonn

			29 is in the sequence since there are no primes in the interval(4*29,4*31)=(116,124)

M. F. Hasler, Table of n, a(n) for n = 1..4911 (all terms up to 5*10^5).

Cf. A195270, A195271, A195377, A195325, A195329, A195379.

Select[Prime[Range[500]],Count[Range[4*#,4*NextPrime[#]],?PrimeQ]==0&] (* _Harvey P. Dale, Jan 05 2013 *)

is_A218561(p)=isprime(p)&nextprime(4*p)>=4*nextprime(p+1)  \\ - M. F. Hasler, Nov 03 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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A218769 Let (p,p+2) be the n-th twin prime pair. a(n) is the least integer r > 1 for which the interval (r*p, r*(p+2)) contains no primes, or a(n)=0, if no such r exists.

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A195379 3.5-gap primes: Primes prime(k) such that there is no prime between 7*prime(k)/2 and 7*prime(k+1)/2.

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A218561 4-gap primes: Prime p is a term iff there is no prime between 4*p and 4*nextprime(p), where nextprime=A151800.

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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.

A218769 Let (p,p+2) be the n-th twin prime pair. a(n) is the least integer r > 1 for which the interval (rp, r(p+2)) contains no primes, or a(n)=0, if no such r exists.

A195379 3.5-gap primes: Primes prime(k) such that there is no prime between 7prime(k)/2 and 7prime(k+1)/2.

A218561 4-gap primes: Prime p is a term iff there is no prime between 4p and 4nextprime(p), where nextprime=A151800.