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Search: id:A159802
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| A159802 |
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Number of primes q with (2m)^2+1 <= q < (2m+1)^2-2m |
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+0
1
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| 1, 2, 2, 2, 4, 2, 2, 4, 2, 3, 4, 4, 4, 4, 4, 5, 4, 7, 6, 8, 5, 4, 7, 7, 6, 9, 7, 7, 6, 8, 7, 9, 7, 10, 11, 7, 10, 12, 9, 6, 9, 8, 8, 8, 9, 8, 10, 10, 12, 11, 11, 12, 13, 9, 12, 14, 13, 11, 10, 14, 11, 14, 15, 12, 16, 14, 16, 11, 12, 11, 12, 14, 14, 15, 15, 13, 17, 15, 16, 18, 17, 15, 12, 12
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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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, primes q with (2m)^2+1 <= q < (2m+1)^2-2m
4) It is conjectured that a(m) >= 1
5) No a(m) with m>1 is known, where a(m)=1
This is a bisection of A089610 and hence related to a conjecture of Opperman. [From T. D. Noe (noe(AT)sspectra.com), Apr 22 2009]
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REFERENCES
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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
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1) m=1: 5 <= q < 7 => prime 5: a(1)=1
2) m=2: 17 <= q < 21 => primes 17, 19: a(2)=2
3) m=3: 37 <= q < 43 => primes 37, 41: a(3)=2
4) m=30: 3601 <= q < 3661 => primes 3607,3613,3617,3623,3631,3637,3643,3659: a(30)=8
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MATHEMATICA
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f[n_] := PrimePi[(2 n + 1)^2 - 2 n - 1] - PrimePi[(2 n)^2]; Table[ f@n, {n, 85}] [From Robert G. Wilson v (rgwv(AT)rgwv.com), May 04 2009]
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CROSSREFS
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A145354, A157884, A014085
Sequence in context: A160691 A049716 A066671 this_sequence A049627 A134058 A086973
Adjacent sequences: A159799 A159800 A159801 this_sequence A159803 A159804 A159805
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KEYWORD
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nonn
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AUTHOR
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Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 22 2009
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EXTENSIONS
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More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), May 04 2009
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