A317990 - OEIS

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

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OFFSET

2,2

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

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

PROG

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

CROSSREFS

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

Sequence in context: A073810 A055255 A057768 * A106030 A104888 A286885

Adjacent sequences: A317987 A317988 A317989 * A317991 A317992 A317993

KEYWORD

nonn

AUTHOR

Jianing Song, Oct 03 2018

EXTENSIONS

Offset corrected by Jianing Song, Mar 31 2019

STATUS

approved