A191483 -id:A191483 - cp's OEIS frontend

A003642 Number of genera of imaginary quadratic field with discriminant -k, k = A191483(n).

1, 1, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 4, 4, 2, 2, 2, 2, 4, 2, 2, 4, 2, 2, 2, 4, 4, 4, 4, 2, 2, 4, 4, 2, 4, 2, 2, 4, 2, 2, 2, 4, 8, 2, 2, 4, 2, 4, 2, 2, 4, 4, 4, 2, 2, 4, 4, 2, 4, 2, 2, 4, 2, 2, 4, 8, 2, 4, 2, 4, 4, 2, 2, 4, 4, 4, 4, 2, 2, 2, 4, 2, 4, 2, 4, 8, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 4, 4, 2, 4, 4, 4, 2, 2, 4

Offset: 1

N. J. A. Sloane, Mira Bernstein

nonn

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, A003641, A191483.

2^(PrimeNu[Select[Range[1000], Mod[#, 4] == 0 && SquareFreeQ[#/4] && Mod[#, 16] != 12&]] - 1) (* Jean-François Alcover, Jul 25 2019, after Andrew Howroyd in A191483 *)

for(n=1, 1000, if(isfundamental(-n) && n%2==0, print1(2^(omega(n) - 1), ", "))) \\ Andrew Howroyd, Jul 24 2018

a(n) = 2^(omega(A191483(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.

A003657 Discriminants of imaginary quadratic fields, negated.

Original entry on oeis.org

3, 4, 7, 8, 11, 15, 19, 20, 23, 24, 31, 35, 39, 40, 43, 47, 51, 52, 55, 56, 59, 67, 68, 71, 79, 83, 84, 87, 88, 91, 95, 103, 104, 107, 111, 115, 116, 119, 120, 123, 127, 131, 132, 136, 139, 143, 148, 151, 152, 155, 159, 163, 164, 167, 168, 179, 183, 184, 187, 191

Offset: 1

N. J. A. Sloane, Mira Bernstein

nonn

Negative of fundamental discriminants D := b^2-4*a*c<0 of definite integer binary quadratic forms F=a*x^2+b*x*y+c*y^2. See Buell reference pp. 223-234. See 4*A089269 = A191483 for even a(n) and A039957 for odd a(n). - Wolfdieter Lang, Nov 07 2003
All prime numbers in the set of the absolute values of negative fundamental discriminants are Gaussian primes (A002145). - Paul Muljadi, Mar 29 2008
Complement: 1, 2, 5, 6, 9, 10, 12, 13, 14, 16, 17, 18, 21, 22, 25, 26, 27, 28, 29, 30, 32, 33, 34, 36, ..., . - Robert G. Wilson v, Jun 04 2011
The asymptotic density of this sequence is 3/Pi^2 (A104141). - Amiram Eldar, Feb 23 2021

Duncan A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989.
Henri Cohen, A Course in Computational Algebraic Number Theory, Springer, 1993, p. 514.
Paulo Ribenboim, Algebraic Numbers, Wiley, NY, 1972, p. 97.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..3000
David Broadhurst and Daniele Dorigoni, Resurgent Lambert series with characters, arXiv:2507.21352 [math.NT], 2025. See pp. 30, 53.
Steven R. Finch, Class number theory. [Cached copy, with permission of the author]
Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.
Eric Weisstein's World of Mathematics, Class Number, Dirichlet L-Series, Fundamental Discriminant.

Cf. A002145, A003658, A039957 (odd terms), A191483 (even terms), A104141.

FundamentalDiscriminantQ[n_Integer] := n != 1 && (Mod[n, 4] == 1 || !Unequal[ Mod[n, 16], 8, 12]) && SquareFreeQ[n/2^IntegerExponent[n, 2]] (* via Eric E. Weisstein *); -Select[-Range@ 194, FundamentalDiscriminantQ] (* Robert G. Wilson v, Jun 01 2011 *)

ok(n)={isfundamental(-n)} \\ Andrew Howroyd, Jul 20 2018

ok(n)={n<>1 && issquarefree(n/2^valuation(n,2)) && (n%4==3 || n%16==8 || n%16==4)} \\ Andrew Howroyd, Jul 20 2018

[n for n in (1..200) if is_fundamental_discriminant(-n)==1] # G. C. Greubel, Mar 01 2019

A039957 Squarefree numbers congruent to 3 mod 4.

Original entry on oeis.org

3, 7, 11, 15, 19, 23, 31, 35, 39, 43, 47, 51, 55, 59, 67, 71, 79, 83, 87, 91, 95, 103, 107, 111, 115, 119, 123, 127, 131, 139, 143, 151, 155, 159, 163, 167, 179, 183, 187, 191, 195, 199, 203, 211, 215, 219, 223, 227, 231, 235, 239, 247, 251, 255

Offset: 1

R. K. Guy

Negatives of odd fundamental discriminants D := b^2-4*a*c<0 of definite integer binary quadratic forms F=a*x^2+b*x*y+c*y^2. See Buell reference pp. 224-230. See 4*A089269 = A191483 for the even case and A003657 for combined even and odd numbers. - Wolfdieter Lang, Nov 07 2003

The asymptotic density of this sequence is 2/Pi^2 = 0.202642... (A185197). - Amiram Eldar, Feb 10 2021

Richard A. Mollin, Quadratics, CRC Press, 1996, Tables B1-B3.
Duncan A. Buell, Binary Quadratic Forms, Springer-Verlag, NY, 1989.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
A. M. Legendre, Diviseurs de la forme t^2+au^2 a étant un nombre de la forme 4n-1, Essai sur la Théorie des Nombres An VI, Table V.

Cf. A039955, A039956, A185197, A191483.

a039957 n = a039957_list !! (n-1)
a039957_list = filter ((== 3) . (`mod` 4)) a005117_list
-- Reinhard Zumkeller, Aug 15 2011

[4*n+3: n in [0..63] | IsSquarefree(4*n+3)];  // Bruno Berselli, Mar 04 2011

fQ[n_] := SquareFreeQ[n] && Mod[n, 4] == 3; Select[ Range@ 258, fQ] (* Robert G. Wilson v, Mar 02 2011 *)
Select[Range[3,300,4],SquareFreeQ] (* Harvey P. Dale, Mar 08 2015 *)

is(n)=n%4==3 && issquarefree(n) \\ Charles R Greathouse IV, Feb 07 2017

Offset corrected

cp's OEIS Frontend

A003642 Number of genera of imaginary quadratic field with discriminant -k, k = A191483(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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A003657 Discriminants of imaginary quadratic fields, negated.

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A039957 Squarefree numbers congruent to 3 mod 4.

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