

A006374


Number of positive definite reduced binary quadratic forms of discriminant 4*n.
(Formerly M0214)


5



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, 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, 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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,3


COMMENTS

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


REFERENCES

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", SpringerVerlag, p. 360.
A. Hurwitz and N. Kritikos, transl. with add. material by W . C. Schulz, "Lectures on Number Theory", SpringerVerlag, New York, 1986, p. 186.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

Table of n, a(n) for n=1..91.


EXAMPLE

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


CROSSREFS

Cf. A006371, A006375, A096446, A096445.
Sequence in context: A229942 A025422 A078640 * A193677 A281855 A137921
Adjacent sequences: A006371 A006372 A006373 * A006375 A006376 A006377


KEYWORD

nonn,nice,easy


AUTHOR

N. J. A. Sloane


EXTENSIONS

Name clarified by Wolfdieter Lang, Mar 31 2019


STATUS

approved



