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A338861 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.

%I A338861 #46 Sep 06 2024 17:27:49
%S A338861 1,2,6,15,42,143,399,1190,4209,10920,37245,109886,339745,1037186,
%T A338861 3205734,9784263,29837784,93313919,289627536
%N A338861 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 A338861 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 A338861 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 A338861 Stuart Anderson, <a href="http://www.squaring.net/sq/sr/spsr/spsr.html">Catalogues of Simple Perfect Squared Rectangles</a>.
%H A338861 Stuart Anderson, <a href="http://www.squaring.net/sq/sr/sisr/sisr.html">Simple Imperfect Squared Rectangles, orders 9 to 24</a>.
%H A338861 Bertram Felgenhauer, <a href="http://int-e.eu/~bf3/squares/">Filling rectangles with integer-sided squares</a>.
%H A338861 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 A338861 Rainer Rosenthal, <a href="/A338861/a338861.png">Rectangle tiled by 19 squares with maximum area a(19)</a>
%e A338861 a(6) = 11*13 = 143.
%e A338861 Dissection of the 11 X 13 rectangle into 6 squares:
%e A338861 .
%e A338861           +-----------+-------------+
%e A338861           |           |             |
%e A338861           |           |             |
%e A338861           |   6 X 6   |    7 X 7    |
%e A338861           |           |             |
%e A338861           |           |             |
%e A338861           +---------+-+             |
%e A338861           |         +-+-----+-------+
%e A338861           |  5 X 5  |       |       |
%e A338861           |         | 4 X 4 | 4 X 4 |
%e A338861           |         |       |       |
%e A338861           +---------+-------+-------+
%e A338861 .
%e A338861 a(19) = 16976*17061 = 289627536.
%e A338861 Dissection of the 16976 X 17061 rectangle into 19 squares:
%e A338861 .
%e A338861        +----------------+-------------+
%e A338861        |                |             |
%e A338861        |                |             |
%e A338861        |                |     7849    |
%e A338861        |      9212      |             |
%e A338861        |                |             |
%e A338861        |                |             |
%e A338861        |                |------+------|
%e A338861        |________________|      |      |
%e A338861        |             |   see   | 4109 |
%e A338861        |             |Rosenthal|      |
%e A338861        |             |  link +-+------+
%e A338861        |     7764    |-------|        |
%e A338861        |             |       |  5018  |
%e A338861        |             | 4279  |        |
%e A338861        |             |       |        |
%e A338861        +-------------+-------+--------+
%e A338861 .
%Y A338861 Cf. A219158, A340726, A340920.
%Y A338861 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.
%K A338861 nonn,hard,more
%O A338861 1,2
%A A338861 _Rainer Rosenthal_, Nov 12 2020
%E A338861 a(11)-a(17) from _Hugo Pfoertner_ based on data from squaring.net website, Nov 17 2020
%E A338861 a(18) from _Hugo Pfoertner_, Feb 18 2021
%E A338861 a(19) from _Hugo Pfoertner_, Apr 02 2021