A317990 Number of genera of real quadratic field Q(sqrt(k)), k squarefree > 1.

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

Offset

2,2

Comments

The number of genera of a quadratic field is equal to the number of elements x in the form class group such that x^2 = e where e is the identity.

This is the analog of A003643 for real quadratic fields.

Not to be confused with A391435, which gives the numbers of elements that square to the identity in the *class groups* of real quadratic fields. - Jianing Song, Dec 09 2025

Links

Jianing Song, Table of n, a(n) for n = 2..10000

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

Formula

For omega(k) the number of distinct prime divisors of k:

- a(n) = 2^(omega(A005117(n)) - 1) if A005117(n) == 1 (mod 4);

- a(n) = 2^(omega(4*A005117(n)) - 1) = 2^(omega(A005117(n)) - 1) if A005117(n) == 2 (mod 4);

- a(n) = 2^(omega(4*A005117(n)) - 1) = 2^(omega(A005117(n))) if A005117(n) == 3 (mod 4). [Corrected by Jianing Song, Dec 10 2025]

Prog

(PARI) for(n=2, 200, if(issquarefree(n), print1(2^(omega(n*if(n%4>1, 4, 1)) - 1), ", ")))

Crossrefs

Cf. A003643 (for imaginary quadratic fields), A005117.

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, this seq. |

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

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

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

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

| 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: A073810 A055255 A057768 * A106030 A104888 A286885

Adjacent sequences: A317987 A317988 A317989 * A317991 A317992 A317993

Keyword

nonn

Author

Jianing Song, Oct 03 2018

Extensions

Offset corrected by Jianing Song, Mar 31 2019

Status

approved