%I #23 Dec 10 2025 06:54:20

%S 40,60,65,85,96,104,105,120,136,140,156,160,165,168,185,192,200,204,

%T 205,220,221,224,232,240,252,260,264,265,273,280,285,288,296,300,305,

%U 312,320,325,336,340,345,348,352,357,360,364,365,377,380,384,385,408

%N Positive discriminants of orders of quadratic fields with class number 2.

%C Not to be confused with A391424, the positive discriminants of orders of quadratic fields with *form* class number 2. - _Jianing Song_, Dec 09 2025

%C Discriminants of orders of real quadratic fields whose form class group quotient by {I,-I} is isomorphic to C_2, 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

%C The fundamental terms are listed in A094619.

%H Jianing Song, <a href="/A344408/b344408.txt">Table of n, a(n) for n = 1..10001</a>

%H Rick L. Shepherd, <a href="http://libres.uncg.edu/ir/uncg/f/Shepherd_uncg_0154M_11099.pdf">Binary quadratic forms and genus theory</a>, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.

%o (PARI) isA344408(d) = (d>0) && !issquare(d) && ((d%4==0)||(d%4==1)) && quadclassunit(d)[2]==[2]

%Y Cf. A322710 (the negative discriminant case), A094619.

%Y Sequences related to the class numbers of real quadratic fields:

%Y | Class numbers | Form class no. |

%Y -------------+---------------+----------------+

%Y Fundamental | 1: A003656 | 1: A003655 |

%Y disc. only | 2: A094619 | 2: A391420 |

%Y (A003658) | 3: A094612 | 3: A391421 |

%Y | List: A003652 | List: A003646 |

%Y -------------+---------------+----------------+

%Y All disc. | 1: A133315 | 1: A391423 |

%Y (A079896) | 2: this seq. | 2: A391424 |

%Y | 3: A344409 | 3: A391425 |

%Y | List: A391418 | List: A087048 |

%Y For a list of sequences related to the class groups of real quadratic fields, see A390079.

%K nonn

%O 1,1

%A _Jianing Song_, May 17 2021

%E Name clarified by _Jianing Song_, Dec 09 2025