Unconditional Proofs of Two Conjectures in OEIS A217564

This paper establishes unconditional proofs of two assertions recorded as conjectures in OEIS A217564. Let \(p_n\) denote the \(n\)-th prime and define \(a(n)=\pi(p_{n+1}/2)-\pi(p_n/2)\). We prove that \(\limsup_{n\to\infty}a(n)=\infty\) and strengthen the conjecture that \(a(n)=0\) infinitely often by establishing a lower asymptotic density of at least \(1/2\). The proofs combine the Maynard--Tao theorem for admissible linear forms, the Chinese remainder theorem, and the prime number theorem. We further generalize both conclusions to \(a_d(n)=\pi(p_{n+1}/d)-\pi(p_n/d)\) for every fixed integer \(d\ge2\), obtaining a lower density of zeros of at least \(1-1/d\). All results are unconditional and require no unproved hypotheses. The arguments derive these consequences from established theorems; originality and priority are not claimed.