For each positive integer k and each primorial Pₙ = p₁ p₂ pₙ, we study the reduced fraction obtained from k/Pₙ after cancellation. We prove that once pₙ exceeds the largest prime divisor of k, the reduction pattern stabilizes: the reduced numerator equals k / rad (k) and the reduced denominator equals Pₙ / rad (k), where rad (k) denotes the radical of k. This stable behavior yields a natural three-way classification of all integers k 2: primes (numerator 1, denominator misses exactly one prime), squarefree composites (numerator 1, denominator misses at least two primes), and non-squarefree integers (numerator >1). We determine the natural densities of the three classes to be 0, 6/², and 1-6/², respectively. The classification is further refined by the stable missing-prime statistic, which equals the number of distinct prime divisors of k on the squarefree locus. We construct the associated Dirichlet generating series---expressed in terms of the Riemann zeta function (s) and the prime zeta function P (s) ---and a bivariate Euler product that uniformly encodes the refined squarefree strata. As a core application, we show that the framework generates a canonical family of unit fractions with squarefree denominators, and we prove that every positive rational with squarefree reduced denominator admits a constructive Egyptian fraction representation using only these building blocks, with an explicit bound on the number of terms. An algebraization of the framework via local valuation profiles and ideal-theoretic extensions to number fields is also initiated.
Jianming Wang (Sat,) studied this question.