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 A159802 Number of primes q with (2m)^2+1 <= q < (2m+1)^2-2m. 0
 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; refs; listen; history; text; internal format)
 OFFSET 1,2 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, 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 Oppermann. - T. D. Noe, Apr 22 2009] 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 LINKS EXAMPLE 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. MATHEMATICA f[n_] := PrimePi[(2 n + 1)^2 - 2 n - 1] - PrimePi[(2 n)^2]; Table[ f@n, {n, 85}] [Robert G. Wilson v, May 04 2009] CROSSREFS Cf. A145354, A157884, A014085. Sequence in context: A049716 A188903 A066671 * A329586 A255336 A292929 Adjacent sequences:  A159799 A159800 A159801 * A159803 A159804 A159805 KEYWORD nonn AUTHOR Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 22 2009 EXTENSIONS More terms from Robert G. Wilson v, May 04 2009 STATUS approved

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Last modified January 19 14:53 EST 2020. Contains 331049 sequences. (Running on oeis4.)