A391424 Positive discriminants of orders of quadratic fields with form class number 2.

12, 21, 24, 28, 32, 33, 40, 44, 45, 48, 56, 57, 65, 69, 72, 76, 77, 80, 84, 85, 88, 92, 93, 104, 108, 112, 117, 124, 128, 129, 132, 133, 141, 152, 153, 161, 172, 176, 177, 180, 184, 185, 188, 189, 200, 201, 208, 209, 213, 216, 217, 228, 232, 236, 237, 245, 248, 249, 253, 260, 261, 265, 268, 272, 276, 284, 296, 297

Offset

1,1

Comments

Not to be confused with A344408, the positive discriminants of orders of quadratic fields with class number 2.

Let h(D) and h+(D) be respectively the class number and the form class number of discriminant D. Since h+(D) = h(D) if the fundamental unit has norm -1 and 2*h(D) otherwise, this sequence consists of D such that either h(D) = 1 and that the fundamental unit has norm 1, or h(D) = 2 and that the fundamental unit has norm -1.

Links

Jianing Song, 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.

Prog

(PARI) QFBclassno(D) = qfbclassno(D) * if (D > 0 && quadunitnorm(D) > 0, 2, 1)
isA391424(n) = n%4 <= 1 && !issquare(n) && QFBclassno(n) == 2 \\ quadunitnorm() requires PARI-GP of version 2.15 or higher

Crossrefs

Cf. A306638 (norms of fundamental units of orders of real quadratic fields).

For a list of sequences related to the class numbers of real quadratic fields, see A087048.

Sequence in context: A116700 A132132 A248378 * A391420 A349649 A327297

Adjacent sequences: A391421 A391422 A391423 * A391425 A391426 A391427

Keyword

nonn,changed

Author

Jianing Song, Dec 09 2025

Status

approved