On the products (1ℓ+1)(2ℓ+1)⋯(nℓ+1)(1ℓ+1)(2ℓ+1)⋯(nℓ+1)

In this note, we prove that (i) for any odd prime power ℓ and any positive integer n  , the product (1ℓ+1)(2ℓ+1)⋯(nℓ+1)(1ℓ+1)(2ℓ+1)⋯(nℓ+1) is not a powerful number; (ii) for any positive odd number ℓ  , there exists an integer NℓNℓ such that for any positive integer n⩾Nℓn⩾Nℓ, the product (1ℓ+1)(2ℓ+1)⋯(nℓ+1)(1ℓ+1)(2ℓ+1)⋯(nℓ+1) is not a powerful number.