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A080359
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The smallest integer x > 0 such that the number of primes in (x/2,x] equals n.
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29
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2, 3, 13, 19, 31, 43, 53, 61, 71, 73, 101, 103, 109, 113, 139, 157, 173, 181, 191, 193, 199, 239, 241, 251, 269, 271, 283, 293, 313, 349, 353, 373, 379, 409, 419, 421, 433, 439, 443, 463, 491, 499, 509, 523, 577, 593, 599, 601, 607, 613, 619, 647, 653, 659
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| a(n) is the same as: Smallest integer x > 0 such that the number of unitary-prime-divisors of x! equals n.
Let p_n be the n-th prime. If p_n>3 is in the sequence, then all integers (p_n-1)/2, (p_n-3)/2, ..., (p_(n-1)+1)/2 are composite numbers. [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Aug 12 2009]
For n>=3, denote by q(n) the prime which is the nearest from the left to a(n)/2. Then there exists a prime between 2q(n) and a(n). Converse, generally speaking, is not true, i.e. there exist primes outside the sequence, but possess such property (e.g., 131) [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Aug 14 2009]
See sequence A164958 for a generalization. [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Sep 02 2009]
a(n) is the n-th Labos prime.
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LINKS
| Daniel Forgues, Table of n, a(n) for n=1..4460
J. Sondow, Ramanujan Prime in MathWorld
V. Shevelev, On critical small intervals containing primes [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Aug 20 2009]
V. Shevelev, Ramanujan and Labos primes, their generalizations and classifications of primes
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FORMULA
| a(n)=Min{x; Pi[x]-Pi[x/2]=n}=Min{x; A056171(x)=n}=Min{x; A056169(n!)=n}; where Pi()=A000720().
a(n)<=A193507(n) (Cf. A194186). - Vladimir Shevelev, Aug 18 2011
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EXAMPLE
| n=5: in 31! five unitary-prime-divisors appear (firstly): {17,19,23,29,31}, while other primes {2,3,5,7,11,13} are at least squared. Thus a(5)=31.
Consider a(9)=71. Then the nearest prime<71/2 is q(9)=31, and between 2q(9) and a(9), i.e. between 62 and 71 there exists a prime (67). [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Aug 14 2009]
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MATHEMATICA
| nn=1000; t=Table[0, {nn+1}]; s=0; Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s<=nn && t[[s+1]]==0, t[[s+1]]=k], {k, Prime[3*nn]}]; Rest[t]
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CROSSREFS
| Cf. A056171, A056169, A000720, A000142.
Cf. A104272 Ramanujan primes.
Cf. A080360 largest integer x with n primes in (x/2,x].
A164554 A164288 A164333 A164294 A164372 A164371 [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Aug 20 2009]
Sequence in context: A118134 A143871 A194598 * A193507 A103087 A135118
Adjacent sequences: A080356 A080357 A080358 * A080360 A080361 A080362
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KEYWORD
| nonn
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AUTHOR
| Labos E. (labos(AT)ana.sote.hu), Feb 21 2003
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EXTENSIONS
| Definition corrected by Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 10 2008
Shrunk title and moved part of title to comments. John W. Nicholson (reddwarf2956(AT)yahoo.com), Sept 18 2011.
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