Powerful numbers in \((1^k +1)(2^k+1) \cdots (n^k +1)\) (Q452395)

Define \(\Omega_k(n)=\prod_{a=1}^n (a^k+1)\). The authors show that if \(k\) is an odd prime, then \(\Omega_k(n)\) is not powerful. For the proof they show by elementary means that if \(p\in[n/2+1, n+1]\) is a prime, satisfying \(p\neq k\) and \(p\not\equiv 1\pmod{k}\), then \(p\mid\Omega_k(n)\), but \(p^2\nmid\Omega_k(n)\). In particular, if such a prime exists, then \(\Omega_k(n)\) is not powerful. They then combine an explicit version of the prime number theorem with the Brun-Titchmarsh inequality to show that for \(n>9\) such a prime always exists.