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%I #14 Mar 06 2024 04:47:03
%S 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,
%T 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,
%U 11,14,15,12,16,14,16,11,12,11,12,14,14,15,15,13,17,15,16,18,17,15,12,12
%N Number of primes q with (2m)^2+1 <= q < (2m+1)^2-2m.
%C 1) Immediate connection to unsolved problem, is there always a prime between n^2 and (n+1)^2 ("full" interval of two consecutive squares).
%C 2) See sequence A145354 and A157884 for more details to this new improved conjecture.
%C 3) First ("left") half interval, primes q with (2m)^2+1 <= q < (2m+1)^2-2m.
%C 4) It is conjectured that a(m) >= 1.
%C 5) No a(m) with m>1 is known, where a(m)=1.
%C This is a bisection of A089610 and hence related to a conjecture of Oppermann. - _T. D. Noe_, Apr 22 2009]
%D L. E. Dickson, History of the Theory of Numbers, Vol, I: Divisibility and Primality, AMS Chelsea Publ., 1999
%D R. K. Guy, Unsolved Problems in Number Theory (2nd ed.) New York: Springer-Verlag, 1994
%D P. Ribenboim, The New Book of Prime Number Records. Springer. 1996
%e 1) m=1: 5 <= q < 7 => prime 5: a(1)=1.
%e 2) m=2: 17 <= q < 21 => primes 17, 19: a(2)=2.
%e 3) m=3: 37 <= q < 43 => primes 37, 41: a(3)=2.
%e 4) m=30: 3601 <= q < 3661 => primes 3607,3613,3617,3623,3631,3637,3643,3659: a(30)=8.
%t 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 *)
%Y Cf. A145354, A157884, A014085.
%K nonn
%O 1,2
%A Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 22 2009
%E More terms from _Robert G. Wilson v_, May 04 2009