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A159804
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Number of primes q with (2m-1)^2+1 <= q < (2m)^2-(2m-1)
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0
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1, 1, 1, 1, 2, 2, 3, 4, 1, 3, 4, 2, 4, 4, 4, 5, 6, 5, 3, 6, 5, 7, 6, 6, 6, 5, 7, 6, 7, 8
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| 1) Immediate connection to unsolved problem, is there always a prime between n^2 and (n+1)^2 ("full" interval of two consecutive squares)
2) See sequence A145354 and A157884 for more details to this new improved conjecture
3) First ("left") half interval: number of primes q with (2m-1)^2+1 <= q < (2m)^2-(2m-1)
4) It is conjectured that a(m) >= 1
5) No a(m) with m>9 is known, where a(m)=1
This is a bisection of A089610 and hence related to a conjecture of Oppermann. [From T. D. Noe (noe(AT)sspectra.com), Apr 22 2009]
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REFERENCES
| L. E. Dickson, History of the Theory of Numbers, Vol, I: Divisibility and Primality, AMS Chelsea Publ., 1999
R. K. Guy, Unsolved Problems in Number Theory (2nd ed.) New York: Springer-Verlag, 1994
P. Ribenboim, The New Book of Prime Number Records. Springer. 1996
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EXAMPLE
| 1) m=1: 2 <= q < 3 => prime 2: a(1)=1
2) m=5: 82 <= q < 91 => primes 83,89: a(5)=2
3) m=9: 290 <= q < 307 => prime 293: a(9)=1
4) m=30: 3482 <= q < 3541 => prime 3491,3499,3511,3517,3527,3529,3533,3539: a(30)=8
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CROSSREFS
| A145354, A157884, A014085
Sequence in context: A071477 A071507 A071509 * A104567 A087824 A008951
Adjacent sequences: A159801 A159802 A159803 * A159805 A159806 A159807
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KEYWORD
| nonn
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AUTHOR
| Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 22 2009
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