A003657 Discriminants of imaginary quadratic fields, negated. (Formerly M2332)

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

1,1

Comments

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

References

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).

Links

T. D. Noe, Table of n, a(n) for n=1..3000

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.

Mathematica

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 *)

Prog

(PARI) ok(n)={isfundamental(-n)} \\ Andrew Howroyd, Jul 20 2018
(PARI) ok(n)={n<>1 && issquarefree(n/2^valuation(n, 2)) && (n%4==3 || n%16==8 || n%16==4)} \\ Andrew Howroyd, Jul 20 2018
(Sage) [n for n in (1..200) if is_fundamental_discriminant(-n)==1] # G. C. Greubel, Mar 01 2019

Crossrefs

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

Sequence in context: A173467 A050122 A179016 * A003644 A196923 A192051

Adjacent sequences: A003654 A003655 A003656 * A003658 A003659 A003660

Keyword

nonn

Author

N. J. A. Sloane, Mira Bernstein

Status

approved