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

%I #9 Feb 05 2019 23:59:59

%S 1,1,1,2,1,2,2,2,4,3,3,4,4,4,4,5,4,5,6,5,6,5,7,7,4,10,5,5,10,8,6,7,5,

%T 7,5,7,10,7,10,12,11,10,7,11,10,10,10,12,8,9,11,9,9,8,9,15,15,9,14,11,

%U 14,17,11,11,10,17,14,15,13,17,17,13,12,16,13,20,17,11,14,14,17,16,17,16

%N Number of primes p with (2m)^2-(2m-1) <= p < (2m)^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)^2-(2m-1) <= p < (2m)^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 Except for a(1), 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 m=1: 3 <= p < 4 => prime 3: a(1)=1;

%e m=4: 57 <= p < 64 => primes 59,61: a(4)=2;

%e m=5: 91 <= p < 100 => prime 97: a(5)=1;

%e m=30: 3541 <= p < 3600 => primes 3541, 3547, 3557, 3559, 3571, 3581, 3583, 3593: a(30)=8.

%p A159805 := proc(n) local a,p; a := 0 ; for p from 4*n^2-2*n+1 to 4*n^2-1 do if isprime(p) then a := a+1 ; fi; od: a ; end: seq(A159805(n),n=1..120) ; # _R. J. Mathar_, Apr 22 2009

%Y Cf. A145354, A157884, A014085.

%K nonn

%O 1,4

%A Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 22 2009

%E More terms from _R. J. Mathar_, Apr 22 2009