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

Author

Jianing Song, Sep 25 2018

Keywords

• nonn

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 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).

Links

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

Crossrefs

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

Real discriminant case: A317991.

Programs

PARI

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

Formula

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