A391440 - OEIS

A391440

Number of genera in the order of real quadratic fields with discriminant A079896(n).

1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 4, 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 4, 1, 1, 2, 4, 2, 1, 2, 1, 1, 2, 4, 2, 1, 2, 2, 2, 2, 2, 1, 4, 2, 2, 1, 1, 2, 2, 4, 1, 4, 2, 1, 4, 4, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 4, 1, 1, 2, 2, 4, 2, 2, 2, 1, 2, 2, 2, 4, 2, 4

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OFFSET

1,3

COMMENTS

Number of elements that square to the identity in the form class group of discriminant D = A079896(n).

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.

FORMULA

For D = A079896(n), a(n) = 2^(omega(|D|) - t), where omega = A001221, and t = 0 if 32|D, t = 2 if D == 4 (mod 16), t = 1 otherwise. See A391441.

PROG

(PARI) r(D) = omega(D) - if(D%32==0, 0, if(D%16==4, 2, 1)) \\ gives 2-rank of Cl+(D)

for(D=1, 1000, if(D%4<=1 && !issquare(D), print1(2^r(D), ", ")))

CROSSREFS

Cf. A003640 (for imaginary quadratic fields).

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: this seq. |

| Exponent <= 2: A391419 | exponent <= 2: A390079 |

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

Sequence in context: A353332 A353362 A256122 * A087048 A109700 A087742

Adjacent sequences: A391437 A391438 A391439 * A391441 A391442 A391443

KEYWORD

nonn

AUTHOR

Jianing Song, Dec 09 2025

STATUS

approved