A037449 Discriminant of quadratic field Q(sqrt(n)).

1, 8, 12, 1, 5, 24, 28, 8, 1, 40, 44, 12, 13, 56, 60, 1, 17, 8, 76, 5, 21, 88, 92, 24, 1, 104, 12, 28, 29, 120, 124, 8, 33, 136, 140, 1, 37, 152, 156, 40, 41, 168, 172, 44, 5, 184, 188, 12, 1, 8, 204, 13, 53, 24, 220, 56, 57, 232, 236, 60, 61, 248, 28, 1, 65, 264, 268, 17, 69

Offset

1,2

Comments

For the discriminant of the quadratic field Q(sqrt(-n)), see A204993.

a(n) is the smallest positive N such that (n/k) = (n/(k mod N)) for every odd k that is coprime to n, where (n/k) is the Jacobi symbol. As we have Dirichlet's theorem on arithmetic progressions, a(n) is also the smallest positive N such that (n/p) = (n/(p mod N)) for every odd prime p that is not a factor of n. - Jianing Song, May 16 2024

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

S. R. Finch, Class number theory

Steven R. Finch, Class number theory [Cached copy, with permission of the author]

Formula

Let b(n) = A007913(n), then a(n) = b(n) if b(n) == 1 (mod 4) and 4*b(n) otherwise. - Jianing Song, May 16 2024

a(n) = 4*n/A389136(n)^2. - Hugo Pfoertner, Sep 24 2025

Mathematica

Table[NumberFieldDiscriminant[Sqrt[n]], {n, 100}] (* Artur Jasinski, Jan 27 2012 *)

Prog

(PARI) vector(150, n, quaddisc(n))
(SageMath)
[fundamental_discriminant(n) for n in (1..69)] # Peter Luschny, Oct 15 2018

Crossrefs

Cf. A007913, A204993, A389136.

Sequence in context: A126814 A172391 A227239 * A070477 A070697 A004473

Adjacent sequences: A037446 A037447 A037448 * A037450 A037451 A037452

Keyword

nonn,easy

Author

Jason Earls, Jun 30 2001

Status

approved