A391425 - OEIS

A391425

Positive discriminants of orders of quadratic fields with form class number 3.

148, 229, 257, 404, 733, 761, 788, 916, 1028, 1076, 1229, 1373, 1396, 1489, 1492, 1556, 1901, 2089, 2213, 2228, 2557, 2677, 2708, 2713, 2777, 2804, 2836, 2857, 2917, 2932, 3028, 3044, 3221, 3229, 3316, 3508, 3877, 3889, 3988, 4001, 4481, 4493, 4597, 4649, 4729, 4852, 4916, 4933

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OFFSET

1,1

COMMENTS

Not to be confused with A344409, the positive discriminants of orders of quadratic fields with class number 3.

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 h(D) = 3 and that the fundamental unit has norm -1.

Note that the form class group has 2-rank omega(|D|) - t, where omega = A001221, and t = 0 if 32|D, t = 2 if D == 4 (mod 16), t = 1 otherwise (see A391441). As a result, this sequence is exactly the terms in A344409 that are of the form 8, p or 4*p for prime p == 1 (mod 4).

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)

isA391425(n) = n%4 <= 1 && !issquare(n) && QFBclassno(n) == 3 \\ quadunitnorm() requires PARI-GP of version 2.15 or higher

CROSSREFS

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

Sequences related to the class numbers of real quadratic fields:

| Class numbers | Form class no. |

-------------+---------------+----------------+

Fundamental | 1: A003656 | 1: A003655 |