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A164368
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Primes p with the property: if q is the smallest prime > p/2, then a prime exists between p and 2q.
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34
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2, 11, 17, 29, 41, 47, 59, 67, 71, 97, 101, 107, 109, 127, 137, 149, 151, 167, 179, 181, 191, 197, 227, 229, 233, 239, 241, 263, 269, 281, 283, 307, 311, 347, 349, 367, 373, 401, 409, 419, 431, 433, 439, 461, 487, 491, 503, 521, 569, 571, 587, 593, 599, 601, 607
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OFFSET
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1,1
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COMMENTS
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The Ramanujan primes possess the following property:
If p = prime(n) > 2, then all numbers (p+1)/2, (p+3)/2, ..., (prime(n+1)-1)/2 are composite.
The sequence equals all primes with this property, whether Ramanujan or not.
All Ramanujan primes A104272 are in the sequence.
Every lesser of twin primes (A001359), beginning with 11, is in the sequence. - Vladimir Shevelev, Aug 31 2009
109 is the first non-Ramanujan prime in this sequence.
A very simple sieve for the generation of the terms is the following: let p_0=1 and, for n>=1, p_n be the n-th prime. Consider consecutive intervals of the form (2p_n, 2p_{n+1}), n=0,1,2,... From every interval containing at least one prime we remove the last one. Then all remaining primes form the sequence. Let us demonstrate this sieve: For p_n=1,2,3,5,7,11,... consider intervals (2,4), (4,6), (6,10), (10,14), (14,22), (22,26), (26,34), ... . Removing from the set of all primes the last prime of each interval, i.e., 3,5,7,13,19,23,31,... we obtain 2,11,17,29, etc. - Vladimir Shevelev, Aug 30 2011
This sequence and A194598 are the mutually wrapping up sequences:
A194598(1) <= a(1) <= A194598(2) <= a(2) <= ...
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 1..1000
V. Shevelev, On critical small intervals containing primes (arXiv:0908.2319v9 [math.NT])
V. Shevelev, Ramanujan and Labos primes, their generalizations and classifications of primes (arXiv:0909.0715v13 [math.NT])
V. Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq. 15 (2012) Article 12.5.4
J. Sondow, Ramanujan Prime in MathWorld
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 (see Section 5 Prime gaps)
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EXAMPLE
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2 is in the sequence, since then q=2, and there is a prime 3 between 2 and 4. - N. J. A. Sloane, Oct 15 2009
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MAPLE
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a:= proc(n) option remember; local q, k, p;
k:= nextprime (`if` (n=1, 1, a(n-1)));
do q:= nextprime (floor (k/2));
p:= nextprime (k);
if p<2*q then break fi;
k:= p
od; k
end:
seq (a(n), n=1..55); # Alois P. Heinz, Aug 30 2011
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MATHEMATICA
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Reap[Do[q=NextPrime[p/2]; If[PrimePi[2*q] != PrimePi[p], Sow[p]], {p, Prime[Range[100]]}]][[2, 1]]
fQ[n_] := PrimePi[ 2NextPrime[n/2]] != PrimePi[n]; Select[ Prime@ Range@ 105, fQ]
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PROG
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(PARI) is(n)=nextprime(n+1)<2*nextprime(n/2) && isprime(n) \\ Charles R Greathouse IV, Apr 24 2015
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CROSSREFS
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Cf. A164332, A164333, A104272, A164288, A080359, A164294, A001262, A001567, A062568, A141232, A193507, A194184, A194186, A194598.
Sequence in context: A066794 A087379 A019364 * A194658 A104272 A214934
Adjacent sequences: A164365 A164366 A164367 * A164369 A164370 A164371
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KEYWORD
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nonn
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AUTHOR
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Vladimir Shevelev, Aug 14 2009
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EXTENSIONS
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Definition clarified and simplified by Jonathan Sondow, Oct 25 2011
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STATUS
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approved
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