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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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%I #14 Jul 13 2021 02:16:57

%S 0,2,4,6,9,11,15,17,23,25,30,34,40,44,50,54,61,65,71,79,87,91,98,104,

%T 114,122,128,138,147,155,161,171,183,193,199,209,217,225,237,249,262,

%U 276,286,296,308,318,331,345,359,365,377,391,410,418,428

%N 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.

%C 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).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EisensteinPrime.html">Eisenstein prime</a>

%H Wikipedia, <a href="https://www.wikipedia.org/wiki/Eisenstein_integer">Eisenstein integer</a>.

%e 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.

%t 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]

%Y Cf. A055667, A062711.

%K nonn

%O 1,2

%A _Frank M Jackson_ and Michael B Rees, Mar 05 2018