A300416 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.

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

Offset

1,2

Comments

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).

Links

Table of n, a(n) for n=1..55.

Eric Weisstein's World of Mathematics, Eisenstein prime

Wikipedia, Eisenstein integer.

Example

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.

Mathematica

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]

Crossrefs

Cf. A055667, A062711.

Sequence in context: A164286 A054519 A168434 * A353134 A038107 A377057

Adjacent sequences: A300413 A300414 A300415 * A300417 A300418 A300419

Keyword

nonn

Author

Frank M Jackson and Michael B Rees, Mar 05 2018

Status

approved