Dynamics of the Digit Sum of Integer Powers: Cycle Structure and Finite Search

We study the arithmetic function f_k(n) = s(n^k), where s(n) denotes the base-10 digit sum and k ≥ 2 is fixed. We analyze its iterative dynamics, focusing on fixed points, cycles of length 2, and higher-order cycles. We prove that all cycles of f_k are contained in a finite set {1, ..., 10^{D_max(k)} − 1}, where D_max(k) is the largest integer D satisfying 10^{D−1} ≤ 9kD. This establishes that the search for cycles is finite. Using this bound, we classify all cycles computationally for 2 ≤ k ≤ 10. The results show that the dynamics are dominated by fixed points and short cycles. This work combines empirical exploration, mathematical formalization, and computational verification. Implementations in Python and Excel are provided for reproducibility. This version is written in Portuguese. An English version will be made available in a subsequent revision.