On the products (13 + q3)(23 + q3)⋯(n3 + q3)

For a positive integer n and a fixed positive integer q, let Cn=∏k=1n(k3+q3). We prove that Cn is not a powerful number if n≥max⁡{q,1198−q}, using some basic estimates on primes in arithmetic progressions. As applications of our main results, we discuss the powerfulness of Cn for 1≤q≤10.