login
A317991
2-rank of the narrow class group of real quadratic field with discriminant A003658(n), n >= 2.
3
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
2,18
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.
FORMULA
a(n) = omega(A003658(n)) - 1 = log_2(A317989(n)), where omega(k) is the number of distinct prime divisors of k.
MATHEMATICA
PrimeNu[Select[Range[2, 300], NumberFieldDiscriminant[Sqrt[#]] == #&]] - 1 (* Jean-François Alcover, Jul 25 2019 *)
PROG
(PARI) for(n=2, 1000, if(isfundamental(n), print1(omega(n) - 1, ", ")))
CROSSREFS
KEYWORD
nonn
AUTHOR
Jianing Song, Oct 03 2018
EXTENSIONS
Offset corrected by Jianing Song, Mar 31 2019
STATUS
approved