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

Author

Jianing Song, Oct 03 2018

Keywords

• nonn

Comments

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

This is the analog of A003640 for real quadratic fields. Note that for this case "the class group" refers to the narrow class group, or the form class group of indefinite binary quadratic forms with discriminant k.

Links

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

Crossrefs

Cf. A003640, A003658, A087048, A317990, A317991.

Programs

Mathematica

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

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), ", ")))

Sage

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

Formula

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

Extensions

Offset corrected by Jianing Song, Mar 31 2019