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A164333
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Primes prime(k) such that all integers in the interval [(prime(k-1)+1)/2, (Prime(k)-1)/2] are composite numbers.
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13
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Let p_k be the k-th prime. A prime p is in the sequence iff the interval of the form (2p_k, 2p_(k+1)), containing p, contains also a prime less than p. The sequence is connected with the following classification of primes: two first primes 2,3 form a separate set of primes; let p>=5 be in interval(2p_k, 2p_(k+1)), then 1)if in this interval there are primes only more than p, then p is called a right prime; 2)if in this interval there are primes only less than p, then p is called a left prime;
3)if in this interval there are prime more 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) more than 2 are either right or central primes; the left primes form sequence A166308, and all Labos primes (A080359) more than 3 are either left or central primes; the central primes form A166252 and the isolated primes form A166251. [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Oct 10 2009]
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LINKS
| V. Shevelev, On critical small intervals containing primes, arXiv:0908.2319
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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)<... [From Vladimir Shevelev, Oct 10 2009]
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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. [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Oct 10 2009]
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MAPLE
| isA164333 := proc(n)
local i ;
if isprime(n) and n > 3 then
for i from (prevprime(n)+1)/2 to (n-1)/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
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CROSSREFS
| Cf. A080359, A104272, A164288, A164294, A164332, A001262, A001567, A062568, A141232
A164368 A164554 A166251 A166252 [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Oct 10 2009]
Sequence in context: A040092 A079130 A085413 * A069324 A040047 A163847
Adjacent sequences: A164330 A164331 A164332 * A164334 A164335 A164336
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KEYWORD
| nonn
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AUTHOR
| Vladimir Shevelev (shevelev(AT)bgu.ac.il), Aug 13 2009
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EXTENSIONS
| In the formula I added \{2,3} Vladimir Shevelev (shevelev(AT)bgu.ac.il), Aug 17 2009
Definition rephrased by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 02 2009
I corrected my comment from 10.10.09 and the first formula (union of...). - Vladimir Shevelev (shevelev(AT)bgu.ac.il), Oct 16 2009
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