A006374 - OEIS

%I M0214 #25 Apr 01 2019 06:26:46

%S 1,1,2,2,2,2,2,3,3,2,4,4,2,4,4,4,4,3,4,6,4,2,6,6,3,6,6,4,6,4,6,7,4,4,

%T 8,8,2,6,8,6,8,4,4,10,6,4,10,8,5,7,8,6,6,8,8,12,4,2,12,8,6,8,10,8,8,8,

%U 4,12,8,4,14,9,4,10,10,10,8,4,10,14,9,4,12,12,4,10,12,6,12,10,8

%N Number of positive definite reduced binary quadratic forms of discriminant -4*n.

%C In Hurwitz and Kritikos (HK) a definite form a*x^2 + 2*b*x*y + c*y^2, denoted by f(a,b,c), has discriminant Delta = -D = b^2 - 4*a*c < 0. Usually this is F = [a,2*b,c] with discriminant Disc = 4*(b^2 - a*c) = 4*Delta. A definite form is reduced if 2*|b| <= a <= c, and if any of the inequalities reduces to an equality then b >= 0 (HK, p. 179). The positive definite case has a > 0 and c  > 0. Here the forms F do not need to satisfy gcd(a,2*b,c) = 1. - _Wolfdieter Lang_, Mar 31 2019

%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 360.

%D A. Hurwitz and N. Kritikos, transl. with add. material by W . C. Schulz, "Lectures on Number Theory", Springer-Verlag, New York, 1986, p. 186.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%e a(5) = 2 because the two forms F = [a,2*b,c] with discriminant Disc = -4*5 = -20 are [1,0,5] and [2,2,3]. ([2,-2,3] is not reduced, (-2,2,-3) is not positive definite). - _Wolfdieter Lang_, Mar 31 2019

%Y Cf. A006371, A006375, A096446, A096445.

%K nonn,nice,easy

%O 1,3

%A _N. J. A. Sloane_

%E Name clarified by _Wolfdieter Lang_, Mar 31 2019