Abstract We study the statistical and dynamical properties of iterates of the sum-of-divisors function (n) via the normalized ratio \ Rq (n) = ^ (q) (n) n, q 3, \ using large-scale computation (n 10⁶), regression modeling, extreme-value analysis, and a finite-difference analogue of Lyapunov diagnostics. Empirically, Rq (n) is strongly right-skewed and heavy-tailed, with rare large spikes linked to highly composite integers; Lyapunov analysis shows a contraction-dominated local sensitivity consistent with boundedness. Regression on arithmetic predictors (log-scale, divisor count, prime factor indicators) explains much central variation but leaves structured extreme residuals, motivating peaks-over-threshold analysis. We introduce an entropy-based lower-tail criterion linking bounded empirical Shannon entropy to exponential bounds on upper-tail mass and proving that bounded entropy with a vanishing-tail condition forces infinitely many n with Rq (n) T. Combined with a fractal-geometry analysis (box--counting dimension D₁₎ₗ 0. 9925) of the integer-dynamic attractor, this yields measurable constraints supporting the Schinzel Conjecture for q 3. Our entropy–fractal framework, supported by reproducible computations, offers a statistically grounded and computationally verified pathway toward resolving this conjecture
Zeraoulia Rafik (Mon,) studied this question.