A317989 -id:A317989 - cp's OEIS frontend

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

Jianing Song, Oct 03 2018

nonn

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.

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

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

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

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

Offset corrected by Jianing Song, Mar 31 2019

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

Original entry on oeis.org

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

Jianing Song, Oct 03 2018

nonn

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.

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

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

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

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

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

Offset corrected by Jianing Song, Mar 31 2019

cp's OEIS Frontend

A317990 Number of genera of real quadratic field Q(sqrt(k)), k squarefree > 1.

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