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%I #12 Feb 06 2019 00:00:11
%S 1,1,2,2,1,3,2,3,4,4,3,5,3,5,4,4,5,2,6,4,4,7,3,8,5,7,6,5,7,8,10,5,8,7,
%T 10,8,7,10,9,7,10,9,13,10,11,11,11,11,11,12,9,9,11,14,12,11,12,12,11,
%U 15,12,11,14,12,12,14,15,12,15,14,17,18,20,18,17,14,18,12,15,15,15,14,21
%N Number of primes p with (2m+1)^2 - 2m <= p < (2m+1)^2.
%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) Second ("right") half interval: number of primes p with (2m+1)^2-2m <= p < (2m+1)^2.
%C 4) It is conjectured that a(m) >= 1.
%C 5) No a(m) with m>5 is known, where a(m)=1.
%C This is a bisection of A094189 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: 7 <= p < 9 => prime 7: a(1)=1.
%e 2) m=2: 21 <= p < 25 => prime 23: a(2)=1.
%e 3) m=3: 43 <= p < 49 => primes 43, 47: a(3)=2.
%e 4) m=30: 3661 <= p < 3721 => primes 3671,3673,3677,3691,3697,3701,3709,3719: a(30)=8.
%p A159803 := proc(n) local a,p; a := 0 ; for p from 4*n^2+2*n+1 to 4*n^2+4*n do if isprime(p) then a := a+1 ; fi; od: a ; end: seq(A159803(n),n=1..120) ; # _R. J. Mathar_, Apr 22 2009
%Y Cf. A145354, A157884, A014085.
%K nonn
%O 1,3
%A Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 22 2009
%E More terms from _R. J. Mathar_, Apr 22 2009
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