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A121243
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Primes of the form 4*x^2 + 4*x*y + 9*y^2.
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1
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17, 73, 89, 97, 193, 233, 241, 281, 401, 433, 449, 601, 617, 641, 673, 769, 929, 937, 977, 1009, 1033, 1049, 1097, 1193, 1289, 1297, 1361, 1409, 1433, 1481, 1489, 1609, 1697, 1721, 1753, 1801, 1873, 1913
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OFFSET
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1,1
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COMMENTS
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This sequence is complementary to A105389 in the sense that the two sequences are disjoint and their union constitutes all primes p satisfying Mod[p,8]=1.
Primes satisfying Mod[p,8]=1 are of form x^2+8y^2 (A007519), with the sequence above as odd y, while A105389 is even y. This can be seen by expressing the former as (2x+y)^2+8y^2 (where y can only be odd), while the latter is u^2+8(2v)^2. [From Tito Piezas III, Jan 01 2009]
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LINKS
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EXAMPLE
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17 = 4*1^2 + 4*1*1 + 9*1^2, 73 = 4*1^2 + 4*1*(-3) + 9*(-3)^2
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MATHEMATICA
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QuadPrimes2[4, -4, 9, 10000] (* see A106856 *)
(* Second program: *)
max = 2000; Table[yy = {y, Floor[-2x/9 - 1/9 Sqrt[9max - 32x^2]], Ceiling[-2x/9 + 1/9 Sqrt[9max - 32x^2]]}; Table[4x^2 + 4 x y + 9y^2, yy // Evaluate], {x, 0, Ceiling[3Sqrt[max]/(4Sqrt[2])]}] // Flatten // Union // Select[#, # <= max && PrimeQ[#]&]& // Quiet (* Jean-François Alcover, Oct 08 2018 *)
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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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