Saturation and Growth of Reachable Prime Quotients

We study the saturation of the prime-input floor-quotient map \[ V(X)=\{ X/p : p≤ X,\ p prime\}, \] with particular emphasis on its prime values and on OEIS A090528. A pointwise theorem of Baker, Harman and Pintz implies complete saturation up to exponent \(19/59\): every integer up to \(X19/59-1\) occurs for all sufficiently large \(X\). Using Jia's exceptional-set theorem for primes in short intervals, we prove that for every fixed \(0<β<19/39\), the missing quotient values up to \(X^β\) form an arbitrarily logarithmically sparse exceptional set, and typical fibers have order of magnitude \(X/(q^2log X)\). As an application, every term of the prime-quotient sequence underlying OEIS A090528 is nonzero, and if \(b(n)\) denotes the prime quotient produced by the least prime denominator for \(n^n\), then \[ n→∞log b(n)/nlog n≥ 19/39. \] We also introduce a complete-saturation exponent \(σ\), prove \(19/59≤σ≤1/2\), and conjecture \(σ=1/2\). The accompanying supporting archive contains the LaTeX source, finite recomputation data, reproducibility scripts, and pure-Python exact certificates for the finite \(n=2,…,51\) alignment.