Unbounded Values and Arbitrarily Long Blocks of Ones in a Prime-Generated Run-Length Sequence

Let \(p_n\) denote the \(n\)-th prime and define \(X_n=\lfloor n^{p_n/n}\rfloor\). OEIS A333138 records a sequence obtained by parsing consecutive nondecreasing blocks of \((X_n)\) according to the convention implemented in the OEIS generator. This paper proves two qualitative properties of that parsed sequence \((b_k)\). First, for every fixed integer \(r\ge1\), there are infinitely many indices \(k\) such that \(b_k\ge r\). Second, for every fixed integer \(K\ge1\), the word \(1^K\) occurs infinitely often in \((b_k)\). Consequently,\[\liminf_{k\to\infty} b_k=1,\qquad \limsup_{k\to\infty} b_k=\infty.\]The proof is based on an exact threshold relating the local behavior of \(n^{p_n/n}\) to consecutive prime gaps, together with floor-safe estimates, the large consecutive prime-gap theorem of Ford–Maynard–Tao, and Maynard’s bounded-diameter prime-cluster results. The paper also formalizes the parsing rule used by OEIS A333138 and provides exact finite verification material in the accompanying supplementary archive.As a concrete finite illustration beyond the currently listed OEIS prefix, the supplementary verifier proves exactly that \(b_{520}=10\), with parser start \(s_{520}=1598\).