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

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 A003643 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 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. A003643, A005117, A087048, A317989, A317992.

Programs

PARI

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

Formula

a(n) = 2^(omega(A005117(n)-1)) = 2^A317992(n), where omega(k) is the number of distinct prime divisors of k.

Extensions

Offset corrected by Jianing Song, Mar 31 2019