A319662 2-rank of the class group of Q(sqrt(-k)), k squarefree.

0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 2, 1, 0, 1, 1, 2, 0, 2, 1, 1, 1, 1, 1, 1, 2, 0, 1, 0, 1, 1, 1, 2, 1, 0, 1, 1, 2, 2, 0, 2, 2, 0, 1, 1, 2, 2, 0, 1, 0, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 0, 3, 1, 0, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 0, 2, 2, 0, 2, 1, 1, 2, 0, 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. A003643).

Links

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

Crossrefs

Cf. A000924, A003643, A005117, A033197, A319659.

Real discriminant case: A317992.

Programs

Mathematica

PrimeNu[#*If[Mod[-#, 4]>1, 4, 1]] - 1& /@ Select[Range[200], SquareFreeQ] (* Jean-François Alcover, Aug 02 2019 *)

PARI

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

Sage

def A319662_list(len):
    L = []
    for n in (1..len):
        if is_squarefree(n):
            if (-n) % 4 > 1: n <<= 2
            L.append(sloane.A001221(n) - 1)
    return L
print(A319662_list(141)) # Peter Luschny, Oct 15 2018

Formula

a(n) = log_2(A003643(n)) = omega(A005117(n)) - 1, where omega(k) is the number of distinct prime divisors of k.