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A159803 Number of primes p with (2m+1)^2 - 2m <= p < (2m+1)^2.

%I A159803 #12 Feb 06 2019 00:00:11
%S A159803 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 A159803 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 A159803 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 A159803 Number of primes p with (2m+1)^2 - 2m <= p < (2m+1)^2.
%C A159803 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 A159803 2) See sequence A145354 and A157884 for more details to this new improved conjecture.
%C A159803 3) Second ("right") half interval: number of primes p with (2m+1)^2-2m <= p < (2m+1)^2.
%C A159803 4) It is conjectured that a(m) >= 1.
%C A159803 5) No a(m) with m>5 is known, where a(m)=1.
%C A159803 This is a bisection of A094189 and hence related to a conjecture of Oppermann. - _T. D. Noe_, Apr 22 2009
%D A159803 L. E. Dickson, History of the Theory of Numbers, Vol, I: Divisibility and Primality, AMS Chelsea Publ., 1999
%D A159803 R. K. Guy, Unsolved Problems in Number Theory (2nd ed.) New York: Springer-Verlag, 1994
%D A159803 P. Ribenboim, The New Book of Prime Number Records. Springer. 1996
%e A159803 1) m=1: 7 <= p < 9 => prime 7: a(1)=1.
%e A159803 2) m=2: 21 <= p < 25 => prime 23: a(2)=1.
%e A159803 3) m=3: 43 <= p < 49 => primes 43, 47: a(3)=2.
%e A159803 4) m=30: 3661 <= p < 3721 => primes 3671,3673,3677,3691,3697,3701,3709,3719: a(30)=8.
%p A159803 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 A159803 Cf. A145354, A157884, A014085.
%K A159803 nonn
%O A159803 1,3
%A A159803 Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 22 2009
%E A159803 More terms from _R. J. Mathar_, Apr 22 2009