A319662 -id:A319662 - cp's OEIS frontend

A319659 2-rank of the class group of imaginary quadratic field with discriminant -k, k = A003657(n).

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

Offset: 1

Jianing Song, Sep 25 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 class group of Q(sqrt(-k)), and #{x belongs to G : x^p = 1} is the number of genera of Q(sqrt(-k)) (cf. A003640).

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

Cf. A003640, A003657, A319660, A319661, A319662.

Real discriminant case: A317991.

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

a(n) = log_2(A003640(n)) = omega(A003657(n)) - 1, where omega(k) is the number of distinct prime divisors of k.

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

Original entry on oeis.org

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

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. A317990).

This is the analog of A319662 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. A005117, A087048, A317990, A317991, A319662.

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

a(n) = omega(A005117(n)) - 1 = log_2(A317990(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

A319659 2-rank of the class group of imaginary quadratic field with discriminant -k, k = A003657(n).

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