

A164368


Primes p with the property: if q is the smallest prime > p/2, then a prime exists between p and 2q.


35



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

1,1


COMMENTS

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.
109 is the first nonRamanujan 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 nth 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:
The sequence is the list of primes p = prime(k) such that there are no primes between prime(k)/2 and prime(k+1)/2. Changing "k" to "k1" and therefore "k+1" to "k" produces a definition very similar to A164333's: it differs by prefixing an initial term 3. From this we get a(n+1) = prevprime(A164333(n)) = A151799(A164333(n)) for n >= 1.
The sequence is the list of primes that are not the largest prime less than 2*prime(k) for any k, so that  as a set  it is the complement relative to A000040 of the set of numbers in A059788.
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LINKS



FORMULA



EXAMPLE

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


MAPLE

a:= proc(n) option remember; local q, k, p;
k:= nextprime(`if`(n=1, 1, a(n1)));
do q:= nextprime(floor(k/2));
p:= nextprime(k);
if p<2*q then break fi;
k:= p
od; k
end:


MATHEMATICA

Reap[Do[q=NextPrime[p/2]; If[PrimePi[2*q] != PrimePi[p], Sow[p]], {p, Prime[Range[100]]}]][[2, 1]]
(* Second program: *)
fQ[n_] := PrimePi[ 2NextPrime[n/2]] != PrimePi[n];
Select[ Prime@ Range@ 105, fQ]


PROG



CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



