A391419 Discriminants of orders of real quadratic fields whose class group is isomorphic to (C_2)^r, r >= 0.

5, 8, 12, 13, 17, 20, 21, 24, 28, 29, 32, 33, 37, 40, 41, 44, 45, 48, 52, 53, 56, 57, 60, 61, 65, 68, 69, 72, 73, 76, 77, 80, 84, 85, 88, 89, 92, 93, 96, 97, 101, 104, 105, 108, 109, 112, 113, 116, 117, 120, 124, 125, 128, 129, 132, 133, 136, 137, 140, 141, 149, 152, 153, 156, 157, 160

Offset

1,1

Comments

Discriminants of orders of real quadratic fields with 1 class per genus in the form class group quotient by {I,-I}, 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)).

Links

Jianing Song, Table of n, a(n) for n = 1..10000

Prog

(PARI) isA341419(n)={n%4<=1 && !issquare(n) && !#select(k->k<>2, quadclassunit(n).cyc)}

Crossrefs

Sequences related to the class groups of real quadratic fields:

| Class groups | Form class groups |

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

Fundamental | 2-rank: A391436, A391437 | 2-rank: A317991, A317992 |

disc. only | # of genera: A391426, A391435 | # of genera: A317989, A317990 |

(A003658) | Exponent <= 2: A391417 | Exponent <= 2: A391422 |

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

All disc. | 2-rank: A391439 | 2-rank: A391441 |

(A079896) | # of genera: A391438 | # of genera: A391440 |

| Exponent <= 2: this sequence | Exponent <= 2: A390079 |

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

Sequence in context: A294227 A116602 A079896 * A390079 A133315 A003658

Adjacent sequences: A391416 A391417 A391418 * A391420 A391421 A391422

Keyword

nonn

Author

Jianing Song, Dec 09 2025

Status

approved