A317989 Number of genera of real quadratic field with discriminant A003658(n), n >= 2.

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

Offset

2,3

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 A003640 for real quadratic fields.

Not to be confused with A391426, 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

a(n) = 2^(omega(A003658(n)-1)) = 2^A317991(n), where omega(k) is the number of distinct prime divisors of k.

Mathematica

2^(PrimeNu[Select[Range[2, 300], NumberFieldDiscriminant[Sqrt[#]]==#&]] - 1) (* Jean-François Alcover, Jul 25 2019 *)

Prog

(PARI) for(n=2, 1000, if(isfundamental(n), print1(2^(omega(n) - 1), ", ")))
(PARI) for(n=2, 1000, if(isfundamental(n), print1(2^#select(t->t%2==0, quadclassunit(n).cyc), ", ")))
(SageMath)
def A317989_list(len):
    L = (sloane.A001221(n) for n in (1..len) if is_fundamental_discriminant(n))
    return [2^(l-1) for l in L]
A317989_list(290) # Peter Luschny, Oct 15 2018

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: this seq., 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: A391419 | Exponent <= 2: A390079 |

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

Sequence in context: A053256 A336498 A102418 * A106032 A003646 A201503

Adjacent sequences: A317986 A317987 A317988 * A317990 A317991 A317992

Keyword

nonn

Author

Jianing Song, Oct 03 2018

Extensions

Offset corrected by Jianing Song, Mar 31 2019

Status

approved