Equality of numerical ranges of matrix powers

For an n-by-n matrix A, we determine when the numerical ranges W(A^k), k≥1, of powers of A are all equal to each other. More precisely, we show that this is the case if and only if A is unitarily similar to a direct sum B⊕C, where B is idempotent and C satisfies W(C^k)⊆W(B)for all k≥1. We then consider, for each n≥1, the smallest integer k_n for which every n-by-n matrix A with W(A)=W(A^k)for all k, 1≤k≤k_n, has an idempotent direct summand. For each n≥1, let p_n be the largest prime less than or equal to n+1. We show that (1)k_n≥p_n for all n, (2)if A is normal of size n, then W(A)=W(A^k)for all k, 1≤k≤p_n, implies A having an idempotent summand, and (3)k₁=2 and k₂=k₃=3. These lead us to ask whether k_n=p_n holds for all n≥1.