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A003657
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Discriminants of imaginary quadratic fields, negated.
(Formerly M2332)
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25
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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 = 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
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REFERENCES
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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).
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LINKS
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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, Jun 01 2011 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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