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A003657 Discriminants of imaginary quadratic fields, negated.
(Formerly M2332)
25
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 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)