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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. 3
1, 2, 6, 15, 42, 143, 399, 1190, 4209, 10920, 37245, 109886, 339745, 1037186, 3205734, 9784263, 29837784, 93313919, 289627536 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

LINKS

Table of n, a(n) for n=1..19.

Stuart Anderson, Catalogues of Simple Perfect Squared Rectangles.

Stuart Anderson, Simple Imperfect Squared Rectangles, orders 9 to 24.

Bertram Felgenhauer, Filling rectangles with integer-sided squares.

MathOverflow, tiling a rectangle with the smallest number of squares, answer by Ed Pegg Jr, Jul 09 2017.

Rainer Rosenthal, Rectangle tiled by 19 squares with maximum area a(19)

EXAMPLE

a(6) = 11*13 = 143.

Dissection of the 11 X 13 rectangle into 6 squares:

.

+-----------+-------------+

| | |

| | |

| 6 X 6 | 7 X 7 |

| | |

| | |

+---------+-+ |

| +-+-----+-------+

| 5 X 5 | | |

| | 4 X 4 | 4 X 4 |

| | | |

+---------+-------+-------+

.

a(19) = 16976*17061 = 289627536.

Dissection of the 16976 X 17061 rectangle into 19 squares:

.

+-------------+-----------+

| | |

| | 7849 |

| 9212 | |

| | |

| |___________|

|_____________| | |

| | see | 4109 |

| |link | |

| 7764 |____++------+

| | | 5018 |

| |4279| |

+------------+----+-------+

.

CROSSREFS

Cf. A219158, A290821 (analog with triangles), A340726, A340920.

Sequence in context: A065178 A178936 A221744 * A340726 A303833 A148438

Adjacent sequences: A338858 A338859 A338860 * A338862 A338863 A338864

KEYWORD

nonn,hard,more

AUTHOR

Rainer Rosenthal, Nov 12 2020

EXTENSIONS

a(11)-a(17) from Hugo Pfoertner based on data from squaring.net website, Nov 17 2020

a(18) from Hugo Pfoertner, Feb 18 2021

a(19) from Hugo Pfoertner, Apr 02 2021

STATUS

approved

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Last modified December 6 10:23 EST 2022. Contains 358630 sequences. (Running on oeis4.)