Prime Witnesses in a Floor-Quotient Sequence

For an integer \(X\ge 1\), let \(\mathcal W(X)\) be the set of primes \(p\) for which \(\lfloor X/p\rfloor\) is prime, and let \(W(X)=|\mathcal W(X)|\). For a fixed prime \(q\), we prove the exact identity\[W_q(X)=\pi(X/q)-\pi(X/(q+1)),\]and hence\[W_q(X)\sim \frac{X}{q(q+1)\log X}.\]A truncation argument gives\[W(X)\sim C\frac{X}{\log X},\qquadC=\sum_{q\ \mathrm{prime}}\frac{1}{q(q+1)}.\] We also determine the limiting distribution of the normalized prime witnesses \(p/X\). A general short-interval transfer principle shows that any exponent \(\theta<1\) for which \([x-x^\theta,x]\) contains a prime for all sufficiently large \(x\) yields a least-witness bound of order \(X^{1/(2-\theta)+\varepsilon}\). Using the theorem of Baker, Harman and Pintz with \(\theta=21/40\) gives the exponent \(40/59\). Applied to \(X=n^n\), these results prove the recorded nonvanishing conjecture for OEIS A090527, whose \(n\)-th term is the least prime \(p\) such that \(\lfloor n^n/p\rfloor\) is prime, and strengthen it quantitatively. Nagura's theorem additionally gives, for every \(n\ge2\), a witness whose quotient is exactly \(2\).