A317992 2-rank of the narrow class group of real quadratic field Q(sqrt(k)), k squarefree > 1.

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

Offset

2,10

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. A317990).

This is the analog of A319662 for real quadratic fields.

Links

Table of n, a(n) for n=2..88.

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

Formula

a(n) = omega(A005117(n)) - 1 = log_2(A317990(n)), where omega(k) is the number of distinct prime divisors of k.

Prog

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

Crossrefs

Cf. A005117, A087048, A317990, A317991, A319662.

Sequence in context: A134541 A286627 A182071 * A228085 A154782 A265196

Adjacent sequences: A317989 A317990 A317991 * A317993 A317994 A317995

Keyword

nonn

Author

Jianing Song, Oct 03 2018

Extensions

Offset corrected by Jianing Song, Mar 31 2019

Status

approved