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A003657
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Discriminants of imaginary quadratic fields, negated.
(Formerly M2332)
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13
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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
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OFFSET
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1,1
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COMMENTS
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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 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 4 2011.
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REFERENCES
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D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989.
H. Cohen, Course in Computational Alg. No. Theory, Springer, 1993, p. 514.
P. 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).
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..3000
S. R. Finch, Class number theory
Eric Weisstein's World of Mathematics, Class Number, Dirichlet L-Series, Fundamental Discriminant
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MATHEMATICA
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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, June 1 2011 *)
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CROSSREFS
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Cf. A002145, Odd terms = A039957, even terms = A191483.
Sequence in context: A173467 A050122 A179016 * A003644 A196923 A192051
Adjacent sequences: A003654 A003655 A003656 * A003658 A003659 A003660
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane, Mira Bernstein
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STATUS
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approved
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