A319661 2-rank of the class group of imaginary quadratic field with discriminant -k, k = A191483(n).

0, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 1, 2, 1, 2, 1, 2, 3, 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. A003642).

Links

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

Crossrefs

Cf. A003642, A191483, A319659, A319660.

Programs

Mathematica

PrimeNu[Select[Range[1000], Mod[#, 4] == 0 && SquareFreeQ[#/4] && Mod[#, 16] != 12&]] - 1 (* Jean-François Alcover, Aug 02 2019, after Andrew Howroyd in A191483 *)

PARI

for(n=1, 1000, if(isfundamental(-n) && n%2==0, print1(omega(n) - 1, ", ")))

Sage

def A319661_list(len):
    L = []
    for n in range(2, len+1, 2):
        if is_fundamental_discriminant(-n):
            L.append(sloane.A001221(n) - 1)
    return L
print(A319661_list(854)) # Peter Luschny, Oct 15 2018

Formula

a(n) = log_2(A003642(n)) = omega(A191483(n)) - 1, where omega(k) is the number of distinct prime divisors of k.