Gaussian primes in the first quadrant, sorted by magnitude and imaginary part.

1, 1, 2, 1, 1, 2, 3, 0, 3, 2, 2, 3, 4, 1, 1, 4, 5, 2, 2, 5, 6, 1, 1, 6, 5, 4, 4, 5, 7, 0, 7, 2, 2, 7, 6, 5, 5, 6, 8, 3, 3, 8, 8, 5, 5, 8, 9, 4, 4, 9, 10, 1, 1, 10, 10, 3, 3, 10, 8, 7, 7, 8, 11, 0, 11, 4, 4, 11, 10, 7, 7, 10, 11, 6, 6, 11, 13, 2, 2, 13, 10, 9, 9, 10

1

Note that after the first Gaussian prime 1+i, all primes are either real (such as 3+0i) or come in pairs (such as 1+2i and 2+i). This sequence is similar to S000409, but with real and imaginary part exchanged for numbers not on the x-axis.

T. D. Noe, Plot of 1232 terms

T. D. Noe, Table of 1232 terms

Eric W. Weisstein, MathWorld: Gaussian Prime

(Mma) nn = 10; t = Select[Flatten[Table[a + b*I, {a, nn}, {b, 0, nn}]], Abs[#] <= nn && PrimeQ[#, GaussianIntegers -> True] &]; t2 = Sort[t, Abs[#1] <= Abs[#2] &]; Flatten[Table[If[Im[z] == 0, {Re[z], 0}, {Im[z], Re[z]}], {z, t2}]]

nonn

T. D. Noe, Dec 15 2014