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

This manuscript studies the iterative dynamics of the arithmetic function f_k(n) = s(n^k), where s(·) denotes the sum of digits in base 10 and k ≥ 2. The paper proves an explicit finite bound for the location of cycles, reducing the cycle search to a finite problem, and reports exact computational classifications for 2 ≤ k ≤ 10. Version 3 (July 2026). This version incorporates corrections and attribution improvements following technical feedback from an independent reader. Summary of changes regarding Version 2: 1. Prior work and attribution (Section 1, Section 5.1, References). The cycle 13 ↔ 16 for k = 2 was already documented in the On-Line Encyclopedia of Integer Sequences: A056527 and A056529 (H. Bottomley, June 2000) and A061905 (A. Auel, 2001). The observation by the first author is now explicitly characterized as an independent rediscovery. Related OEIS records for fixed points of n ↦ s(n^k) (A046000, A061209, A061210, in the circle of ideas of generalized Dudeney numbers) are also cited. The contribution of the paper is restated accordingly: the unified dynamical formulation, the finite-localization theorem, and the complete classification of cycles for 2 ≤ k ≤ 10. 2. Proof of Theorem 0 (Section 4). A new Lemma 4 (absorbing invariant set B_k = max(N_k, max_{n ≤ N_k} f_k(n))) closes a gap in the previous argument: Proposition 2 alone does not guarantee that an orbit remains in {1,…,N_k} after entering it (for k = 2, the minimal N_2 = 17 satisfies f_2(17) = 19 > 17). The statement of Theorem 0 is unchanged. The proof of Theorem 1 is now explicitly independent of Theorem 0. 3. New Corollary 1 (Section 4). Every cycle of f_k is contained in {1,…,9k·D_max(k)}, a sharper bound than the one in Theorem 1; this is the bound used by the Python routine of Appendix B, and Section 6 now states the relationship between the two search ranges. 4. Domain convention (Section 3). ℕ = {1, 2, 3, …} is now fixed explicitly; 0 is excluded from the domain (it would only add the trivial fixed point f_k(0) = 0, unreachable from n ≥ 1). 5. Computational methodology (Section 6.1 and Appendix A). The Python routine of Appendix B (exact arbitrary-precision arithmetic) is now the canonical basis for all results in Table 1. The Excel implementation is repositioned as a didactic illustration with an explicit precision warning: digit sums of powers n^k ≥ 10^15 are unreliable in Excel (15 significant digits; e.g., 99^8 = 9227446944279201, digit sum 72, is represented as 9227446944279200, digit sum 71), affecting parts of the ranges for k = 6, 7, 8; moreover, the spreadsheet formulas detect only fixed points and 2-cycles. Table 1 itself is unchanged: all entries were re-verified with exact integer arithmetic. 6. Figures 1–3 (Section 5). All three basin-of-attraction figures were regenerated programmatically from the exact map f_2 (nodes and edges computed, not drawn by hand). Figure 2 now includes 21; Figure 3 now includes 50, 70 and 97; an incorrect edge 40 → 25 in the previous Figure 3 was corrected to 40 → 7 (since s(40²) = s(1600) = 7). The three figures now display all 99 numbers, as stated in the text. 7. Growth condition for generalizations (Sections 7 and 8). The phrase "g of sufficiently rapid growth" was replaced by the correct hypothesis: the number of decimal digits of g(n) must be o(n) (equivalently, log₁₀ g(n) = o(n)). A counterexample is recorded: g(n) = 10ⁿ − 1 grows exponentially, yet s(g(n)) = 9n, so every orbit of s ∘ g is strictly increasing and never periodic. 8. Acknowledgments. A new section thanks an attentive independent reader whose observations motivated these corrections. No other content was changed.