A003643 -id:A003643 - cp's OEIS frontend

A003640 Number of genera of imaginary quadratic field with discriminant -k, k = A003657(n).

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 2, 1, 1, 1, 4, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 4, 2, 1, 1, 4, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 1, 4, 1, 2, 2, 2, 1, 4, 1, 2, 1, 2, 2, 2, 1, 1, 4, 4, 2, 2, 1, 2, 2, 2, 1, 4, 2, 4, 1, 4, 2, 1, 4, 4, 1, 2, 2, 2, 2, 2, 2, 2, 1, 4, 1, 4, 2, 2, 2, 2, 1, 2

Offset: 1

N. J. A. Sloane, Mira Bernstein

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. - Jianing Song, Jul 24 2018

D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 224-241.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.
Index entries for sequences related to quadratic fields

Cf. A001221 (omega), A003641, A003642, A003643, A003657, A191408.

okQ[n_] := n != 1 && SquareFreeQ[n/2^IntegerExponent[n, 2]] && (Mod[n, 4] == 3 || Mod[n, 16] == 8 || Mod[n, 16] == 4);
Reap[For[n = 1, n <= 1000, n++, If[okQ[n], Sow[2^(PrimeNu[n]-1)]]]][[2, 1]] (* Jean-François Alcover, Aug 16 2019, after Andrew Howroyd *)

for(n=1, 1000, if(isfundamental(-n), print1(2^(omega(n) - 1), ", "))) \\ Andrew Howroyd, Jul 24 2018

for(n=1, 1000, if(isfundamental(-n), print1(2^#select(t->t%2==0, quadclassunit(-n).cyc), ", "))) \\ Andrew Howroyd, Jul 24 2018

a(n) = 2^(t-1) where t = number of distinct prime divisors of A003657(n).

a(n) = 2^(omega(A003657(n)) - 1).

Name clarified and offset corrected by Jianing Song, Jul 24 2018

A317990 Number of genera of real quadratic field Q(sqrt(k)), k squarefree > 1.

Original entry on oeis.org

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

Offset: 2

Jianing Song, Oct 03 2018

nonn

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.

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

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

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

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

Offset corrected by Jianing Song, Mar 31 2019

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

Original entry on oeis.org

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

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. A003643).

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

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

Real discriminant case: A317992.

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

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

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

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

cp's OEIS Frontend

A003640 Number of genera of imaginary quadratic field with discriminant -k, k = A003657(n).

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A317990 Number of genera of real quadratic field Q(sqrt(k)), k squarefree > 1.

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

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