

A164333


Primes prime(k) such that all integers in the interval [(prime(k1)+1)/2, (prime(k)1)/2] are composite numbers.


15



13, 19, 31, 43, 53, 61, 71, 73, 101, 103, 109, 113, 131, 139, 151, 157, 173, 181, 191, 193, 199, 229, 233, 239, 241, 251, 269, 271, 283, 293, 311, 313, 349, 353, 373, 379, 409, 419, 421, 433, 439, 443, 463, 491, 499, 509, 523, 571, 577, 593, 599, 601, 607, 613, 619, 643
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OFFSET

1,1


COMMENTS

Let p_k be the kth prime. A prime p is in the sequence iff the interval of the form (2p_k, 2p_(k+1)), containing p, also contains a prime less than p. 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 the interval (2p_k, 2p_(k+1)), then 1) if in this interval there are only primes greater than p, then p is called a right prime; 2) if in this interval there are only primes less than p, then p is called a left prime; 3) if in this interval there are primes both greater 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) greater than 2 are either right or central primes; the left primes form sequence A166308, and all Labos primes (A080359) greater than 3 are either left or central primes; the central primes form A166252 and the isolated primes form A166251. [Vladimir Shevelev, Oct 10 2009]


LINKS

Table of n, a(n) for n=1..56.
V. Shevelev, On critical small intervals containing primes, arXiv:0908.2319 [math.NT], 2009.
V. Shevelev, Ramanujan and Labos Primes, Their Generalizations, and Classifications of Primes, J. Int. Seq. 15 (2012) # 12.5.4
J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, arXiv:1105.2249 [math.NT], 2011; J. Integer Seq. 14 (2011) Article 11.6.2.


FORMULA

{A080359} union {A164294} = {this sequence} union {2,3}.  Vladimir Shevelev, Oct 29 2011
A164368(2)<a(1) < A164368(3)<a(2) < A164368(4)<a(3)<... [Vladimir Shevelev, Oct 10 2009]


EXAMPLE

Let p=53. We see that 2*23<53<2*29. Since the interval (46, 58) contains prime 47<53 and does not contain any prime more than 53, then, by the considered classification 53 is left prime and it is in the sequence. [Vladimir Shevelev, Oct 10 2009]


MAPLE

isA164333 := proc(n)
local i ;
if isprime(n) and n > 3 then
for i from (prevprime(n)+1)/2 to (n1)/2 do
if isprime(i) then
return false;
end if;
end do;
return true;
else
false;
end if;
end proc:
for i from 2 to 700 do
if isA164333(i) then
printf("%d, ", i);
end if;
end do: # R. J. Mathar, Oct 29 2011


MATHEMATICA

kmax = 200; Select[Table[{(Prime[k  1] + 1)/2, (Prime[k]  1)/2}, {k, 3, kmax}], AllTrue[Range[#[[1]], #[[2]]], CompositeQ]&][[All, 2]]*2 + 1 (* JeanFrançois Alcover, Nov 14 2017 *)


CROSSREFS

Cf. A080359, A104272, A164288, A164294, A164332, A001262, A001567, A062568, A141232.
Cf. A164368, A164554, A166251, A166252. [Vladimir Shevelev, Oct 10 2009]
Sequence in context: A085413 A342943 A244311 * A182365 A069324 A040047
Adjacent sequences: A164330 A164331 A164332 * A164334 A164335 A164336


KEYWORD

nonn


AUTHOR

Vladimir Shevelev, Aug 13 2009


EXTENSIONS

Definition rephrased by R. J. Mathar, Oct 02 2009


STATUS

approved



