The difference between the real and imaginary parts of S000921.
0, 1, 1, 3, 5, 5, 5, 7, 5, 5, 5, 5, 7, 7, 9, 9, 7, 9, 7, 5, 5, 5, 5, 5, 7, 9, 11, 9, 9, 7, 5, 5, 7, 11, 13, 13, 15, 15, 13, 13, 13, 15, 17, 17, 19, 17, 19, 15, 15, 13, 11, 11, 13, 11, 11, 15, 17, 17, 15, 13, 13, 13, 9, 7, 9, 11, 13, 15, 17, 17, 15, 15, 17, 15, 15
1
It appears that the points stay relatively close the the x = y line.
T. D. Noe, Plot of 2000 terms
T. D. Noe, Table of 2000 terms
(Mma) nn = 10; pts = 1000; d = Sort[Select[Flatten[Table[{Sqrt[i^2 + j^2], i, j}, {i, 0, nn}, {j, 0, nn}], 1], #[[1]] <= nn &], #1[[1]] < #2[[1]] &]; p = 1 + I; t = {p}; Do[i = 2; While[i <= Length[d] && ! PrimeQ[p + d[[i, 2]] + I*d[[i, 3]], GaussianIntegers -> True], i++]; If[i >= Length[d], Return[], p = p + d[[i, 2]] + I*d[[i, 3]]; AppendTo[t, p]], {pts - 1}]; Re[t] - Im[t]
Cf. S000921.
nonn
T. D. Noe, Jul 25 2016