A089269 -id:A089269 - cp's OEIS frontend

A006643 Class number of quadratic field with discriminant -4n as n runs through A089269: squarefree numbers congruent to 1 or 2 mod 4.

1, 1, 2, 2, 2, 2, 4, 4, 4, 2, 6, 6, 4, 4, 4, 2, 6, 8, 4, 4, 6, 4, 2, 6, 8, 8, 8, 8, 4, 4, 10, 8, 4, 4, 4, 10, 12, 4, 8, 4, 14, 4, 8, 6, 6, 12, 8, 8, 6, 10, 12, 4, 4, 14, 8, 8, 8, 4, 8, 16, 14, 8, 6, 8, 16, 8, 10, 12, 14, 12, 4, 8, 10, 12, 16, 12, 4, 4, 20, 10, 12, 6, 8, 20, 20, 8, 8, 6, 8, 10, 16

Offset: 1

N. J. A. Sloane, Mira Bernstein

nonn

Equivalently, number of classes of primitive positive definite binary quadratic forms of discriminant -4n as n runs through A089269: squarefree numbers congruent to 1 or 2 mod 4.

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).

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Cf. A006642, A089269, A014599.

A subsequence of A000003.

for (n=1,50, if(issquarefree(n) && (n%4 == 1 || n%4 == 2), print(n, " ", qfbclassno(-4*n)))) \\ N. J. A. Sloane, May 28 2014

Extended and definition corrected by Max Alekseyev, Apr 16 2010

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

A191483 Even discriminants of imaginary quadratic fields, negated.

Original entry on oeis.org

4, 8, 20, 24, 40, 52, 56, 68, 84, 88, 104, 116, 120, 132, 136, 148, 152, 164, 168, 184, 212, 228, 232, 244, 248, 260, 264, 276, 280, 292, 296, 308, 312, 328, 340, 344, 356, 372, 376, 388, 404, 408, 420, 424, 436, 440, 452, 456, 472, 488, 516, 520, 532, 536, 548

Offset: 1

Robert G. Wilson v, Jun 04 2011

Terms of A003657 that are not in A039957.

The asymptotic density of this sequence is 1/Pi^2 (A092742). - Amiram Eldar, Feb 23 2021

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A003657, A039957, A089269, A092742.

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@550, FundamentalDiscriminantQ@# && EvenQ@# &]
(* Second program: *)
Select[Range[600], Mod[#, 4] == 0 && SquareFreeQ[#/4] && Mod[#, 16] != 12&] (* Jean-François Alcover, Jul 25 2019, after Andrew Howroyd *)

ok(n)={isfundamental(-n) && n%2==0} \\ Andrew Howroyd, Jul 25 2018

ok(n)={n%4==0 && issquarefree(n/4) && n%16<>12} \\ Andrew Howroyd, Jul 25 2018

Complement(A003657, A039957).

a(n) = 4*A089269(n). - Andrew Howroyd, Jul 25 2018

cp's OEIS Frontend

A006643 Class number of quadratic field with discriminant -4n as n runs through A089269: squarefree numbers congruent to 1 or 2 mod 4.

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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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A191483 Even discriminants of imaginary quadratic fields, negated.

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