A006371 - OEIS

A006371

Number of positive definite reduced binary quadratic forms of discriminant -A014601(n).
(Formerly M0207)

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

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OFFSET

1,6

REFERENCES

H. Davenport, The Higher Arithmetic. Cambridge Univ. Press, 5th edition, 1982, p. 144.

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

LINKS

Robin Visser, Table of n, a(n) for n = 1..10000

Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.

FORMULA

a(2*n) = A006374(n) for all n > 0. - Robin Visser, May 29 2025

EXAMPLE

From Robin Visser, May 29 2025: (Start)

For n = 6, the a(6) = 2 positive definite reduced binary quadratic forms of discriminant -A014601(6) = -12 are x^2 + 3*y^2 and 2*x^2 + 2*x*y + 2*y^2.

For n = 7, the a(7) = 2 positive definite reduced binary quadratic forms of discriminant -A014601(7) = -15 are x^2 + x*y + 4*y^2 and 2*x^2 + x*y + 2*y^2.

For n = 8, the a(8) = 2 positive definite reduced binary quadratic forms of discriminant -A014601(8) = -16 are x^2 + 4*y^2 and 2*x^2 + 2*y^2. (End)

PROG

(SageMath)

def a(n):

D, ans = 2*n+1-(n+1)%2, 0

for b in range(-isqrt(D/3), isqrt(D/3)+1):

if ((D+b^2)%4 != 0): continue

for a in Integer((D+b^2)/4).divisors():

if ((abs(b)==a) or (a^2==(D+b^2)/4)) and (b < 0): continue

if (a >= abs(b)) and (a^2 <= (D+b^2)/4): ans += 1

return ans # Robin Visser, May 29 2025

CROSSREFS