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A338861
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a(n) is the largest area of a rectangle which can be dissected into n squares with integer sides s_i, i = 1 .. n, and gcd(s_1,...,s_n) = 1.
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3
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1, 2, 6, 15, 42, 143, 399, 1190, 4209, 10920, 37245, 109886, 339745, 1037186, 3205734, 9784263, 29837784, 93313919, 289627536
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OFFSET
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1,2
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COMMENTS
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A219158 gives the minimum number of squares to tile an i x j rectangle. a(n) is found by checking all rectangles (i,j) for which A219158 has a dissection into n squares.
Due to the potential counterexamples to the minimal squaring conjecture (see MathOverflow link), terms after a(19) have to be considered only as lower bounds: a(20) >= 876696755, a(21) >= 2735106696. - Hugo Pfoertner, Nov 17 2020, Apr 02 2021
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LINKS
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EXAMPLE
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a(6) = 11*13 = 143.
Dissection of the 11 X 13 rectangle into 6 squares:
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+-----------+-------------+
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| 6 X 6 | 7 X 7 |
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| | |
+---------+-+ |
| +-+-----+-------+
| 5 X 5 | | |
| | 4 X 4 | 4 X 4 |
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+---------+-------+-------+
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a(19) = 16976*17061 = 289627536.
Dissection of the 16976 X 17061 rectangle into 19 squares:
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+----------------+-------------+
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| | 7849 |
| 9212 | |
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| |------+------|
|________________| | |
| | see | 4109 |
| |Rosenthal| |
| | link +-+------+
| 7764 |-------| |
| | | 5018 |
| | 4279 | |
| | | |
+-------------+-------+--------+
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CROSSREFS
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This sequence and A089047 are effectively analogs for dissecting (or tiling) rectangles and squares respectively. Analogs using equilateral triangular tiles are A014529 and A290821 respectively.
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KEYWORD
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nonn,hard,more,changed
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AUTHOR
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EXTENSIONS
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a(11)-a(17) from Hugo Pfoertner based on data from squaring.net website, Nov 17 2020
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STATUS
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approved
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