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.

1, 2, 6, 15, 42, 143, 399, 1190, 4209, 10920, 37245, 109886, 339745, 1037186, 3205734, 9784263, 29837784, 93313919, 289627536

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 |
       |             |Rosenthal|      |
       |             |  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