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

Author

Rainer Rosenthal, Nov 12 2020

Keywords

• nonn
• hard
• more

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

Examples

			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  |        |
       |             |       |        |
       +-------------+-------+--------+
.
		

Links

• 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)

Crossrefs

Cf. A219158, A340726, A340920.

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.

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