Prime Witnesses in a Floor-Quotient Sequence

For an integer \(X≥ 1\), let \( W(X)\) be the set of primes \(p\) for which \( X/p\) is prime, and let \(W(X)=| W(X)|\). For a fixed prime \(q\), we prove the exact identity\[W_q(X)=π(X/q)-π(X/(q+1)),\]and hence\[W_q(X)~ X/q(q+1)log X.\]A truncation argument gives\[W(X)~ CX/log X,=∑q\ prime1/q(q+1).\] We also determine the limiting distribution of the normalized prime witnesses \(p/X\). A general short-interval transfer principle shows that any exponent \(θ<1\) for which \([x-x^θ,x]\) contains a prime for all sufficiently large \(x\) yields a least-witness bound of order \(X1/(2-θ)+ε\). Using the theorem of Baker, Harman and Pintz with \(θ=21/40\) 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 \( n^n/p\) is prime, and strengthen it quantitatively. Nagura's theorem additionally gives, for every \(n≥2\), a witness whose quotient is exactly \(2\).