A344409 Positive discriminants of orders of quadratic fields with class number 3.

148, 229, 257, 316, 321, 404, 469, 473, 564, 568, 592, 621, 733, 756, 761, 788, 837, 892, 916, 993, 1016, 1028, 1076, 1101, 1229, 1257, 1264, 1284, 1304, 1332, 1373, 1396, 1436, 1489, 1492, 1509, 1524, 1556, 1573, 1593, 1616, 1620, 1772, 1876, 1892, 1901, 1929, 1944

Offset

1,1

Comments

Not to be confused with A391425, the positive discriminants of orders of quadratic fields with *form* class number 3. - Jianing Song, Dec 09 2025

Discriminants of orders of real quadratic fields whose form class group quotient by {I,-I} is isomorphic to C_3, where I is the principal class. (So I corresponds to the form x^2 - (D/4)*y^2 for 4|D and x^2 - x*y - ((D-1)/4)*y^2 for D == 1 (mod 4), and -I corresponds to the form (D/4)*x^2 - y^2 for 4|D and ((D-1)/4)*x^2 - x*y - y^2 for D == 1 (mod 4)). - Jianing Song, Dec 09 2025

The fundamental terms are listed in A094612.

It seems that for most k in this sequence, 4*k is also in this sequence. The smallest k such that this is not true is k = 564.

Conjecture: if a term k is congruent to 4 modulo 16, then k/4 is either here or in A133315; if a term k is congruent to 0 modulo 16, then k/4 is in this sequence.

Conjecture: a term k is in A006832 if and only if k/4 is not in this sequence.

Links

Jianing Song, Table of n, a(n) for n = 1..10001

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

Prog

(PARI) isA344409(d) = (d>0) && !issquare(d) && ((d%4==0)||(d%4==1)) && quadclassunit(d)[2]==[3]

Crossrefs

Cf. A328825 (the negative discriminant case), A094612, A006832.

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

Sequence in context: A061154 A252690 A134212 * A391425 A127028 A392518

Adjacent sequences: A344406 A344407 A344408 * A344410 A344411 A344412

Keyword

nonn

Author

Jianing Song, May 17 2021

Extensions

Name clarified by Jianing Song, Dec 09 2025

Status

approved