A003642 -id:A003642 - 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

A003641 Number of genera of imaginary quadratic field with discriminant -k, k = A039957(n).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Mira Bernstein

nonn

In other words, this is the number of genera of those imaginary quadratic fields that have a discriminant which is odd and fundamental. The discriminant will be squarefree and of the form -4n+1. - Andrew Howroyd, Jul 25 2018

			a(4) = 2 because -15 = -A039957(4) and the number of genera of the quadratic field with discriminant -15 is 2. - _Andrew Howroyd_, Jul 25 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).

Index entries for sequences related to quadratic fields

Cf. A001221 (omega), A003640, A003642, A039957.

2^(PrimeNu[Select[Range[1000], Mod[#, 4] == 3 && SquareFreeQ[#]&]] - 1) (* Jean-François Alcover, Jul 25 2019, after Andrew Howroyd *)

for(n=1, 1000, if(n%4==3 && issquarefree(n), print1(2^(omega(n) - 1), ", "))) \\ Andrew Howroyd, Jul 24 2018

a(n) = 2^(omega(A039957(n)) - 1). - Jianing Song, Jul 24 2018

Name clarified by Jianing Song, Jul 24 2018

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

Original entry on oeis.org

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

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. A003642).

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

Cf. A003642, A191483, A319659, A319660.

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

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

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

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

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

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A003641 Number of genera of imaginary quadratic field with discriminant -k, k = A039957(n).

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

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