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A159805 Number of primes p with (2m)^2-(2m-1) <= p < (2m)^2.

%I A159805 #11 Jun 02 2025 01:38:43
%S A159805 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 A159805 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 A159805 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 A159805 Number of primes p with (2m)^2-(2m-1) <= p < (2m)^2.
%C A159805 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 A159805 2) See sequence A145354 and A157884 for more details to this new improved conjecture.
%C A159805 3) Second ("right") half-interval: number of primes p with (2m)^2-(2m-1) <= p < (2m)^2.
%C A159805 4) It is conjectured that a(m) >= 1.
%C A159805 5) No a(m) with m > 5 is known, where a(m)=1.
%C A159805 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 A159805 L. E. Dickson, History of the Theory of Numbers, Vol, I: Divisibility and Primality, AMS Chelsea Publ., 1999
%D A159805 R. K. Guy, Unsolved Problems in Number Theory (2nd ed.) New York: Springer-Verlag, 1994
%D A159805 P. Ribenboim, The New Book of Prime Number Records. Springer, 1996
%e A159805 m=1: 3 <= p < 4 => prime 3: a(1)=1;
%e A159805 m=4: 57 <= p < 64 => primes 59,61: a(4)=2;
%e A159805 m=5: 91 <= p < 100 => prime 97: a(5)=1;
%e A159805 m=30: 3541 <= p < 3600 => primes 3541, 3547, 3557, 3559, 3571, 3581, 3583, 3593: a(30)=8.
%p A159805 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 A159805 Cf. A145354, A157884, A014085.
%K A159805 nonn
%O A159805 1,4
%A A159805 Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 22 2009
%E A159805 More terms from _R. J. Mathar_, Apr 22 2009