A317991 2-rank of the narrow class group of real quadratic field with discriminant A003658(n), n >= 2.

0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 2, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 2, 0, 0, 2, 1, 1, 1, 1, 0, 2, 1, 1, 0, 1, 2, 0, 1, 2, 2, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 2, 1, 1, 1, 1, 2, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 0, 2, 0, 2, 0, 1

Offset: 2

Author

Jianing Song, Oct 03 2018

Keywords

• nonn

Comments

The p-rank of a finite abelian group G is equal to log_p(#{x belongs to G : x^p = 1}) where p is a prime number. In this case, G is the narrow class group of Q(sqrt(k)) or the form class group of indefinite binary quadratic forms with discriminant k, and #{x belongs to G : x^p = 1} is the number of genera of Q(sqrt(k)) (cf. A317989).

This is the analog of A319659 for real quadratic fields.

Links

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

Crossrefs

Cf. A003658, A087048, A317989, A317992, A319659.

Programs

Mathematica

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

PARI

for(n=2, 1000, if(isfundamental(n), print1(omega(n) - 1, ", ")))

Formula

a(n) = omega(A003658(n)) - 1 = log_2(A317989(n)), where omega(k) is the number of distinct prime divisors of k.

Extensions

Offset corrected by Jianing Song, Mar 31 2019