A319659 -id:A319659 - cp's OEIS frontend

A317991 2-rank of the narrow class group of real quadratic field with discriminant A003658(n), n >= 2.

0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 2, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 2, 0, 0, 2, 1, 1, 1, 1, 0, 2, 1, 1, 0, 1, 2, 0, 1, 2, 2, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 2, 1, 1, 1, 1, 2, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 0, 2, 0, 2, 0, 1

Offset: 2

Jianing Song, Oct 03 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 narrow class group of Q(sqrt(k)) or the form class group of indefinite binary quadratic forms with discriminant k, and #{x belongs to G : x^p = 1} is the number of genera of Q(sqrt(k)) (cf. A317989).

This is the analog of A319659 for real quadratic fields.

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

Cf. A003658, A087048, A317989, A317992, A319659.

PrimeNu[Select[Range[2, 300], NumberFieldDiscriminant[Sqrt[#]] == #&]] - 1 (* Jean-François Alcover, Jul 25 2019 *)

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

a(n) = omega(A003658(n)) - 1 = log_2(A317989(n)), where omega(k) is the number of distinct prime divisors of k.

Offset corrected by Jianing Song, Mar 31 2019

A319660 2-rank of the class group of imaginary quadratic field with discriminant -k, k = A039957(n).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 2, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 1, 0, 2, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 2, 1, 1, 1, 1, 0

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. A003641).

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

Cf. A003641, A039957, A319659, A319661.

for(n=1, 1000, if(n%4==3 && issquarefree(n), print1(omega(n) - 1, ", ")))

a(n) = log_2(A003641(n)) = omega(A039957(n)) - 1, where omega(k) is the number of distinct prime divisors of k.

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.

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

A317991 2-rank of the narrow class group of real quadratic field with discriminant A003658(n), n >= 2.

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A319660 2-rank of the class group 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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A319662 2-rank of the class group of Q(sqrt(-k)), k squarefree.

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Sage

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