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A141123
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Primes of the form -x^2+2*x*y+2*y^2 (as well as of the form 3*x^2+6*x*y+2*y^2).
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43
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2, 3, 11, 23, 47, 59, 71, 83, 107, 131, 167, 179, 191, 227, 239, 251, 263, 311, 347, 359, 383, 419, 431, 443, 467, 479, 491, 503, 563, 587, 599, 647, 659, 683, 719, 743, 827, 839, 863, 887, 911, 947, 971, 983, 1019, 1031, 1091, 1103, 1151, 1163, 1187, 1223
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OFFSET
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1,1
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COMMENTS
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Discriminant = 12. Class = 2. Binary quadratic forms a*x^2 + b*x*y + c*y^2 have discriminant d = b^2 - 4ac and gcd(a, b, c) = 1.
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REFERENCES
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Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.
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LINKS
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Robert Israel, Table of n, a(n) for n = 1..6514
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
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EXAMPLE
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a(3) = 11 because we can write 11 = -1^2 + 2*1*2 + 2*2^2 (or 11 = 3*1^2 + 6*1*1 + 2*1^2).
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MAPLE
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N:= 2000:
S:= NULL:
for xx from 1 to floor(2*sqrt(N/3)) do
for yy from ceil(sqrt(max(1, 3*xx^2-N))) to floor(sqrt(3)*xx) do
S:= S, 3*xx^2-yy^2;
od od:
sort(convert(select(isprime, {S}), list)); # Robert Israel, Jul 20 2020
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MATHEMATICA
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Reap[For[p = 2, p < 2000, p = NextPrime[p], If[FindInstance[p == -x^2 + 2*x*y + 2*y^2, {x, y}, Integers, 1] =!= {}, Print[p]; Sow[p]]]][[2, 1]]
(* or: *)
Select[Prime[Range[200]], # == 2 || # == 3 || Mod[#, 12] == 11&] (* Jean-François Alcover, Oct 25 2016, updated Oct 29 2016 *)
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CROSSREFS
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Cf. A068228 (d = 12), A068231 (Primes congruent to 11 (mod 12)), A141111, A141112 (d = 65).
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
Cf. A084917.
Sequence in context: A158017 A091310 A040994 * A263729 A246496 A119641
Adjacent sequences: A141120 A141121 A141122 * A141124 A141125 A141126
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KEYWORD
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nonn
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AUTHOR
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Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 05 2008
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EXTENSIONS
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More terms from Colin Barker, Apr 05 2015
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STATUS
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approved
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