Schinzel's conjecture predicts that, for any fixed k ≥ 1, the k-fold iterate of the sum-of-divisors function σ, , satisfies lim infn→∞ Rk(n) ∞. While settled for k=1,2, the case k ≥ 3 remains open. We prove a rigorous polylogarithmic upper bound for almost all integers: using Brun's sieve, Dickman–de Bruijn smooth-number estimates, and a refined Turán--Kubilius inequality. Moreover, we quantify the exceptional set, showing that for any fixed , proving that large deviations are extremely rare.We further introduce an entropy-based framework that explains the exponential suppression of the upper tail of Rk(n) and confirm our theoretical predictions with extensive numerical evidence up to n ≤ 1010. These results provide strong new evidence toward Schinzel's conjecture.
Rafik et al. (Fri,) studied this question.