Unconditional Proofs of Two Prime-Counting Conjectures in OEIS A217864

This note establishes unconditionally two prime-counting assertions recorded as conjectures in OEIS A217864. Let \(a(n)\) denote the number of primes in the interval \([\lfloor n\log n\rfloor,(n+1)\log(n+1)]\). We prove that \(\limsup_{n\to\infty}a(n)=\infty\) and that \(a(n)=0\) for infinitely many positive integers \(n\). The first result follows from Maynard's theorem on bounded intervals containing arbitrarily many primes, while the second follows from Westzynthius's theorem on arbitrarily large normalized prime gaps. We also establish a generalization showing that any sequence of consecutive intervals whose lengths tend to infinity contains unbounded numbers of primes. Both conclusions are unconditional and require neither the Riemann Hypothesis nor any other unproved conjecture. The results are deductions from established theorems rather than claims of new prime-gap estimates.