The points given by specific values of k follow the pattern shown in the following snippet, were k=13 is used for a clear example.
Here is the graph with just the points from k=13:
Here are the points with k=4 and their graph: (I think there are less unique points because with k=4 you have points made from sin of multiples of 2 Pi, which give repeated points.)
{{Point[{-1 - Sqrt[2], 0}], Point[{-1, -Sqrt[2]}],
Point[{-1, Sqrt[2]}], Point[{-1 + Sqrt[2], 0}]}}
k =( An odd number ^n) gives points that make up the RHS unit circle (or one close to it) with concentric circles of less and less points, while points that make up the LHS unit circle (or one close to it) with concentric circles of less and less points are given by k =( an even number ^n):
A remarkable quality of the points is that the sum of all k=(positive integer, b)^(positive integer, n), is simply b^n !!!!!!!
Table[k = b^n;
N[Total[Flatten[
Evaluate[
ReIm[(-1)^
k ((Flatten[Evaluate[Block[{x}, Solve[x^k == k, x]]]])[[All,
2]] - 1)]]]]], {n, 2, 7}, {b, 2, 7}] // TableForm
gives
A huge list of all such points
followed by