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The goal is to tile rectangles as small as possible with the given heptomino, in this case number 6 of the 108 heptominoes. We allow the addition of copies of a rectangle. For each rectangle $a\times b$, find the smallest area larger rectangle that copies of $a\times b$ plus at least one of the given heptomino will tile.
Example with the $1\times 1$ you can tile a $2\times 5$ as follows:
Now we don't need to consider $1\times 1$ further as we have found the smallest rectangle tilable with copies of the heptomino plus copies of $1\times 1$.
I found 14 more. I considered component rectangles of width 1 through 11 and length to 31 but my search was not complete.
List of known sizes:
- Width 1: Lengths 1 to 8, 10 to 12
- Width 2: Lengths 2, 3, 5
- Width 3: Length 5
Most of these could be tiled by hand using logic rather than just trial and error.