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%I #41 Sep 29 2023 03:56:17
%S 1,2,6,15,42,143,399,1190,4209,10920,37245,109886,339745,1037186,
%T 3205734,9784263,29837784,93313919,289627536
%N 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.
%C 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.
%C 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
%H Stuart Anderson, <a href="http://www.squaring.net/sq/sr/spsr/spsr.html">Catalogues of Simple Perfect Squared Rectangles</a>.
%H Stuart Anderson, <a href="http://www.squaring.net/sq/sr/sisr/sisr.html">Simple Imperfect Squared Rectangles, orders 9 to 24</a>.
%H Bertram Felgenhauer, <a href="http://int-e.eu/~bf3/squares/">Filling rectangles with integer-sided squares</a>.
%H MathOverflow, <a href="https://mathoverflow.net/questions/116382/tiling-a-rectangle-with-the-smallest-number-of-squares/">tiling a rectangle with the smallest number of squares</a>, answer by Ed Pegg Jr, Jul 09 2017.
%H Rainer Rosenthal, <a href="/A338861/a338861.png">Rectangle tiled by 19 squares with maximum area a(19)</a>
%e a(6) = 11*13 = 143.
%e Dissection of the 11 X 13 rectangle into 6 squares:
%e .
%e +-----------+-------------+
%e | | |
%e | | |
%e | 6 X 6 | 7 X 7 |
%e | | |
%e | | |
%e +---------+-+ |
%e | +-+-----+-------+
%e | 5 X 5 | | |
%e | | 4 X 4 | 4 X 4 |
%e | | | |
%e +---------+-------+-------+
%e .
%e a(19) = 16976*17061 = 289627536.
%e Dissection of the 16976 X 17061 rectangle into 19 squares:
%e .
%e +----------------+-------------+
%e | | |
%e | | |
%e | | 7849 |
%e | 9212 | |
%e | | |
%e | | |
%e | |------+------|
%e |________________| | |
%e | | see | 4109 |
%e | |Rosenthal| |
%e | | link +-+------+
%e | 7764 |-------| |
%e | | | 5018 |
%e | | 4279 | |
%e | | | |
%e +-------------+-------+--------+
%e .
%Y Cf. A219158, A290821 (analog with triangles), A340726, A340920.
%K nonn,hard,more
%O 1,2
%A _Rainer Rosenthal_, Nov 12 2020
%E a(11)-a(17) from _Hugo Pfoertner_ based on data from squaring.net website, Nov 17 2020
%E a(18) from _Hugo Pfoertner_, Feb 18 2021
%E a(19) from _Hugo Pfoertner_, Apr 02 2021
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