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A300416
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Number of prime Eisenstein integers z = x - y*w^2 with |z| <= n and where w = -1/2 + i*sqrt(3)/2 is a primitive cube root of unity.
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0
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0, 2, 4, 6, 9, 11, 15, 17, 23, 25, 30, 34, 40, 44, 50, 54, 61, 65, 71, 79, 87, 91, 98, 104, 114, 122, 128, 138, 147, 155, 161, 171, 183, 193, 199, 209, 217, 225, 237, 249, 262, 276, 286, 296, 308, 318, 331, 345, 359, 365, 377, 391, 410, 418, 428
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OFFSET
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1,2
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COMMENTS
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Two prime Eisenstein integers are not counted separately if they are associated, i.e., if their quotient is a unit (1, -w^2, w, -1, w^2 or -w).
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LINKS
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EXAMPLE
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a(7)=15 because the Eisenstein primes whose modulus <= 7 are 1-w^2, 1-2w^2, 1-3w^2, 1-5w^2, 1-6w^2, 2, 2-w^2, 2-3w^2, 3-w^2, 3-2w^2, 3-4w^2, 4-3w^2, 5, 5-w^2, 6-w^2.
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MATHEMATICA
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a[n_] := Module[{w2=-1/2-I*Sqrt[3]/2, lst={}, x, y, z, Nz}, Do[z=x-w2*y; Nz=x^2+x*y+y^2; If[y==0&&Mod[Sqrt[Nz], 3]==2&&Sqrt[Nz]<=n&&PrimeQ[Sqrt[Nz]], AppendTo[lst, {x, y}], If[Mod[Nz, 3]!=2&&Sqrt[Nz]<=n&&PrimeQ[Nz], AppendTo[lst, {x, y}]]], {x, 0, n}, {y, 0, n}]; Length@lst]; Array[a, 100]
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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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