A005670 -id:A005670 - cp's OEIS frontend

A128909 3D version of A005670. The problem is to dissect an n X n X n cube into smaller integer cubes, the gcd of whose sides is 1, using the smallest number of cubes. The gcd condition exclude dissecting a 6 X 6 X 6 cube into eight 3 X 3 X 3 cubes.

1, 8, 20, 15, 50, 27, 71, 22, 39, 57, 125, 34

Offset: 1

Mauro Fiorentini, Apr 23 2007

As far as I know, no term, (except trivial cases) has been proved optimal. Repeated dissection, as in the above example, shows that if the side is a composite number mn, a(mn) <= a(m) + a(n) - 1. It is an open problem to find a number mn for which a(mn) < a(m) + a(n) - 1. Dissecting a cube with side n into a cube with side n - 1 and several unit cubes gives a trivial bound: a(n) <= 3n^2 - 3n + 2. Dissecting a cube with side n = 2k + 1 into a cube with side k + 1, 7 with side k and several unit cubes gives another trivial bound: a(n) <= (9n^2 - 12n + 31) / 4.

			a(4)=15 because a 4 X 4 X 4 cube can be dissected into 8 2 X 2 X 2, one of which can be dissected into 8 1 X 1 X 1.

Ainley, Stephen, Mathematical Puzzles, Prentice Hall, New York, 1983. p. 81.

Cf. A005670.

A089047 Edge length of largest square dissectable into up to n squares in Mrs. Perkins's quilt problem.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 13, 17, 23, 29, 41, 53, 70, 91, 126, 158, 216, 276, 386, 488, 675, 866, 1179, 1544, 2136, 2739, 3755, 4988, 6443

Offset: 1

R. K. Guy, Dec 03 2003

An inverse to A005670.
More precisely, a(n) = greatest k such that A005670(k) <= n. - Peter Munn, Mar 13 2018
It is not clear which terms have been proved to be correct and which are just conjectures. - Geoffrey H. Morley, Sep 07 2012; N. J. A. Sloane, Jul 06 2017
Terms up to and including a(18) have been proved correct by Ed Wynn (2013). - Stuart E Anderson, Sep 16 2013
A089046 and A089047 are almost certainly correct up to 5000. - Ed Pegg Jr, Jul 06 2017
Deleted terms above 5000. - N. J. A. Sloane, Jul 06 2017
Further best known terms are 8568, 11357, 14877, 19594, 26697, 34632. - Ed Pegg Jr, Jul 06 2017
A290821 is the equivalent sequence for equilateral triangles. - Peter Munn, Mar 06 2018

Ed Pegg, Jr., Mrs. Perkins's Quilts
Ed Pegg Jr. and Richard K. Guy, Mrs. Perkins's Quilts (Wolfram Demonstrations Project)
Ed Wynn, Exhaustive generation of Mrs Perkins's quilt square dissections for low orders, arXiv:1308.5420 [math.CO], 2013-2014.

Cf. A005670, A014529, A018835, A089046, A211302, A290821.

More terms from Ed Pegg Jr, Dec 03 2003
Corrected and extended by Ed Pegg Jr, Apr 18 2010
Duplicate a(6) deleted and a(22)-a(26) revised (from Ed Pegg Jr, Jun 15 2010) by Geoffrey H. Morley, Sep 07 2012
Conjectured terms have been extended up to a(44), based on simple squared square enumeration, by Duijvestijn, Skinner, Anderson, Pegg, Johnson, Milla and Williams. - Stuart E Anderson, Sep 16 2013
a(33) and further terms added by Ed Pegg Jr, Jul 06 2017
Name edited by Peter Munn, Mar 14 2018

A342558 a(n) is the maximum number of distinct currents > 0 in a network of n one-ohm resistors with a total resistance of 1 ohm.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 7, 9, 10, 12, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68

Offset: 1

Hugo Pfoertner and Rainer Rosenthal, May 26 2021

nonn

The resistor networks considered here correspond to multigraphs in which each edge is replaced by one or more one-ohm resistors, and in which there are two distinguished nodes, called poles, between which there is a total resistance of 1 ohm.
It was known that the smallest resistor network with all currents being distinct consists of 21 resistors, found by Duijvestin in 1978. This assumes that the network is planar and thus the analogy to the perfectly tiled squares exists, see A014530. For history and references see link to Stuart Anderson's website "SPSS, Order 21".
In 1983, A. Augusteijn and A. J. W. Duijvestijn  described networks in which the number of resistors in a network with distinct resistances was reduced to 20 by allowing the tiled square to be wrapped onto a cylinder. (see links to their publication and to Stuart Anderson's website "Simple Perfect Square-Cylinders")
For values of n greater than 21 increasingly numerous square divisions with a(n) = n exist so that a(n) = n holds for all n > 21 (see A006983).
In the present sequence, networks based on non-planar graphs are allowed, which makes it possible to find networks with a(n) = n also for n = 18 and n = 19.
In the range from n = 13 to n = 17, larger numbers of distinct currents are found than are possible with the methods for generating Mrs. Perkins's quilts, which naturally correspond to planar graphs.

			Examples for n <= 21 are given in the Pfoertner links. Visualizations of tilings corresponding to optimal networks for n <= 12 are given in the Mathworld "Mrs. Perkins's Quilt" link.

Stuart Anderson, Simple Perfect Squared Squares (SPSSs), Order 21 to 37 and higher orders.
Stuart Anderson, SPSS, Order 21.
Stuart Anderson, Simple Perfect Square-Cylinders (SPSCs); Order 20.
Stuart Anderson, Mrs Perkins's Quilt.
A. Augusteijn and A. J. W. Duijvestijn, Simple perfect square-cylinders of low order, Journal of Combinatorial Theory, Series B, Volume 35, Issue 3, December 1983, Pages 333-337
A. J. W. Duijvestijn, Simple perfect squared square of lowest order, Journal of Combinatorial Theory, Series B, Volume 25, Issue 2, October 1978, Pages 240-243
Ed Pegg Jr., List of solutions for the Mrs. Perkins's Quilt Square packing problem.
Hugo Pfoertner, Examples of networks of n one-ohm-resistors with total resistance of 1 ohm, maximizing the number of distinct currents through the single resistors, May 2021, Jan-Apr 2023
Hugo Pfoertner, Visualization of the resistor networks with n >= 13 using the Mathematica graph function, Jan-Apr 2023
Rainer Rosenthal, Cascade graphs for examples n <= 21, January 2023
Rainer Rosenthal, Network and semi-quilt visualizing a(18), January 2023
Eric Weisstein's World of Mathematics, Mrs. Perkins's Quilt.

Cf. A002962, A005670, A006983, A014530, A160911, A180414, A181340, A217156, A337517, A338593, A342556, A360030.

a(n) = n for n >= 18.

A089046 Least edge-length of a square dissectable into at least n squares in the Mrs. Perkins's quilt problem.

Original entry on oeis.org

1, 2, 2, 2, 3, 3, 4, 5, 6, 8, 10, 14, 18, 24, 30, 40, 54, 71, 92, 121, 155, 210, 266, 360, 476, 642, 833, 1117, 1485, 1967, 2595, 3465, 4534, 5995

Offset: 1

R. K. Guy, Dec 03 2003

An inverse to A005670.
More precisely, a(n) = least k such that A005670(k) >= n. - Peter Munn, Mar 14 2018
It is not clear which terms have been proved to be correct and which are just conjectures. - Geoffrey H. Morley, Aug 29 2012; N. J. A. Sloane, Jul 06 2017
n <= 15 (and possibly 16) proved minimal by J. H. Conway (Conway, J. H. "Re: [math-fun] Mrs. Perkins Quilt - Orders 89, 90 improved over UPIG." math-fun mailing list. October 10, 2003.). The conjectures are best currently known values of a(n) for n > 16. - Stuart E Anderson, Apr 21 2013
A089046 and A089047 are almost certainly correct up to 5000. - Ed Pegg Jr, Jul 06 2017
Deleted terms above 5000. - N. J. A. Sloane, Jul 06 2017
Upper bounds for the next terms in the sequence (which may well be the true values) are 7907, 10293, 13505, 17785, 23239, 31035, 39571, ... - Ed Pegg Jr, Jul 06 2017

H. T. Croft, K. J. Falconer, and R. K. Guy, Section C3 in Unsolved Problems in Geometry, New York: Springer, 1991.
M. Gardner, "Mrs. Perkins's Quilt and Other Square-Packing Problems," Mathematical Carnival, New York: Vintage, 1977.

Stuart E. Anderson, Mrs Perkins's Quilts
J. H. Conway, Mrs. Perkins's Quilt, Proc. Cambridge Phil. Soc. 60, 363-368, 1964.
Ed Pegg, Jr., Mrs. Perkin's Quilts
Ed Pegg Jr., Richard K. Guy, Mrs. Perkins's Quilts (Wolfram Demonstrations Project)
Ed Pegg Jr., Richard K. Guy, Mrs. Perkins's Quilts Notebook source code
G. B. Trustrum, Mrs. Perkins's Quilt, Proc. Cambridge Phil. Soc. 61, 7-11, 1965.
Eric W. Weisstein's World of Mathematics, Mrs. Perkins's Quilt
Ed Wynn, Exhaustive generation of Mrs Perkins's quilt square dissections for low orders, arXiv:1308.5420 [math.CO], 2013-2014.

Cf. A005670, A089046, A089047.

More terms from Ed Pegg Jr, Dec 03 2003
Corrected and extended by Ed Pegg Jr, Apr 18 2010
a(24)-a(27) (from Ed Pegg Jr, Jun 15 2010) added by Geoffrey H. Morley, Aug 29 2012
a(28)-a(30) from Stuart E Anderson, Nov 22 2012
Confirmed a(30) as best known, added a(31) as best known. - Stuart E Anderson, Apr 21 2013
Using James Williams recent discoveries of 15 million simple perfect squared squares in orders 31 to 44 I was able to extend the sequence of best currently known values for optimal quilts from a(32) to a(44). - Stuart E Anderson, Apr 21 2013
Using Anderson and Milla's enumeration of order 31 and 32 perfect squared squares, improved conjectures for a(32) and a(33) were obtained - Stuart E Anderson, Sep 16 2013
a(1)-a(19) confirmed by Ed Wynn, 2013. - N. J. A. Sloane, Nov 29 2013
a(29) corrected and further terms added by Ed Pegg Jr, Jul 06 2017

A160911 a(n) is the number of arrangements of n square tiles with coprime sides in a rectangular frame, counting reflected, rotated or rearranged tilings only once.

Original entry on oeis.org

1, 1, 2, 5, 11, 29, 84, 267, 921, 3481, 14322, 62306, 285845, 1362662, 6681508, 33483830

Offset: 1

Kevin Johnston, Feb 11 2016

There is only one arrangement of 1 square tile: a 1 X 1 rectangle. There is also only 1 arrangement of 2 square tiles: a 2 X 1 rectangle. There are 2 arrangements of 3 square tiles: a 3 X 1 rectangle (three 1 X 1 tiles) and a 3 X 2 rectangle (a 2 X 2 tile and two 1 X 1 tiles).
Short notation for the 2 possible 3-tile solutions:
  3 X 1: 1,1,1
  3 X 2: 2,1,1
More examples see below.
The smallest tile is not always a unit tile, e.g., one of the solutions for 5 tiles is: 6 X 5: 3,3,2,2,2.
My definition of a unique solution is the "signature" string in this notation: the rectangle size for nonsquares and the list of coprime tile sizes sorted largest to smallest. Rotations and reflections of a known solution are not new solutions; rearrangements of the same size tiles within the same overall boundary are not new solutions. But reorganizations of the same size tiles in different boundaries are unique solutions, such as 4 X 1: 1,1,1,1 and 2 X 2: 1,1,1,1.
From Rainer Rosenthal, Dec 23 2022: (Start)
The above description can be abbreviated as follows:
a(n) is the number of (2+n)-tuples (p X q: t_1,...,t_n) of positive integers, such that:
0. p >= q.
1. gcd(t_1,...,t_n) = 1 and t_i >= t_j for i < j and Sum_{i=1..n} t_i^2 = p * q.
2. Any p X q matrix is the disjoint union of contiguous t_i X t_i minors, i = 1..n. (For contiguous minors resp. submatrices see comments in A350237.)
.
The rectangle size p X q may have gcd(p,q) > 1, as seen in the examples for 3 X 2 and 6 X 4. Therefore a(n) >= A210517(n) for all n, and a(6) > A210517(6).
(End)

			From _Rainer Rosenthal_, Dec 24 2022, updated May 09 2024: (Start)
.
                                 |A|
     |A B|                       |B|
     |C D|  (2 X 2: 1,1,1,1)     |C|    (4 X 1: 1,1,1,1)
                                 |D|
.
                                 |A A|
    |A A A|                      |A A|
    |A A A|                      |B B|
    |A A A| (4 X 3: 3,1,1,1)     |B B|  (5 X 2: 2,2,1,1)
    |B C D|                      |C D|
.
    |A A A|
    |A A A|  <=================   3 X 3 minor A
    |A A A|                       2 X 2 minor B
    |B B C|  (5 X 3: 3,2,1,1)     1 X 1 minor C
    |B B D|                       1 X 1 minor D
  ________________________________________________________
       a(4) = 5 illustrated as (p X q: t_1,t_2,t_3,t_4)
         and as p X q matrices with t_i X t_i minors
.
Example configurations for a(6) = 29:
.
                                    |A A A A|
                                    |A A A A|
                                    |A A A A|
      |A A B|         |A B|         |A A A A|
      |A A C|         |C D|         |B B C D|
      |D E F|         |E F|         |B B E F|
   ______________________________________________
      (3 X 3:        (3 X 2:         (6 X 4:
    2,1,1,1,1,1)   1,1,1,1,1,1)    4,2,1,1,1,1)
.                                       _________________________
      |A A A A A A B B B B B B B|      |           |             |
      |A A A A A A B B B B B B B|      |           |             |
      |A A A A A A B B B B B B B|      |    6      |             |
      |A A A A A A B B B B B B B|      |           |      7      |
      |A A A A A A B B B B B B B|      |           |             |
      |A A A A A A B B B B B B B|      |___________|             |
      |C C C C C D B B B B B B B|      |         |1|_____________|
      |C C C C C E E E E F F F F|      |         |       |       |
      |C C C C C E E E E F F F F|      |    5    |  4    |  4    |
      |C C C C C E E E E F F F F|      |         |       |       |
      |C C C C C E E E E F F F F|      |_________|_______|_______|
     _____________________________    _____________________________
         (13 X 11: 7,6,5,4,4,1)           (13 X 11: 7,6,5,4,4,1)
         [rotated by 90 degrees]         [alternate visualization]
.(End)

See A002839 and A217156 for further references and links.

Stuart E. Anderson, Perfect Squared Rectangles and Squared Squares

Cf. A002839, A005670, A113881, A210517, A217156, A219924, A221843, A221844, A221845, A340726, A342558, A350237.

a(15)-a(16) from Kevin Johnston, Feb 11 2016

Title changed from Rainer Rosenthal, Dec 28 2022

A279317 Minimal number of squares in a dissection of an (n) X (n+1) oblong into squares.

Original entry on oeis.org

2, 3, 4, 5, 5, 5, 7, 7, 6, 6, 7, 7, 7, 7, 7, 8, 8, 7, 9, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 8, 9, 9, 9, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 9, 9, 10, 9, 10, 9, 10, 10, 10, 10, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 10, 10, 10, 10, 10, 10, 10, 11, 11, 10, 10, 10, 10, 10, 10, 11, 10, 11, 10, 11, 10, 11, 11, 11, 11, 10, 11, 11, 11, 11, 11, 11, 11, 11, 10, 11, 11, 11, 11, 11, 11, 11, 11, 12

Offset: 1

Ed Pegg Jr, Dec 09 2016

This is very close to b(n) = round(n^(1/3)) + 6. b(18)-a(18) = 2. b(387)-a(387) = 0. All b(n)-a(n) terms in between these points are -1, 0, 1.
Bouwkamp codes of dissections that are believed to be optimal follow.
105 104 60 45 19 26 44 16 12 7 33 28
177 176 99 78 21 57 77 43 16 41 34 9 25
308 307 165 143 22 67 54 142 45 13 41 97 28 69
552 551 312 240 44 60 136 28 16 76 239 101 37 175 138
970 969 546 424 172 252 423 73 50 23 119 80 96 39 293 254
1699 1698 951 748 307 441 747 127 77 50 27 200 134 177 66 509 443
2926 2925 1633 1293 213 299 781 127 86 41 344 1292 509 206 138 68 851 783
5211 5210 2846 2365 571 518 1276 2364 392 90 53 465 302 412 694 584 293 1569 1278
8731 8730 4741 3990 751 1195 2044 3989 1059 444 790 849 884 175 709 256 197 2696 453 2046
15131 15130 8169 6962 2415 4547 6961 1208 1943 1680 263 965 452 1504 702 1378 3621 802 865 2306 2243
25679 25678 13719 11960 1456 1866 2626 6012 303 743 410 11959 1623 440 1516 760 1183 3386 4322 1692 7706 6014
49583 49582 27252 22331 4763 5036 12532 158 4332 273 22330 5080 5309 906 2176 1250 4716 1270 4372 2187 3446 14719 12534

			Oblong 18 X 19 uses 7 squares of size 3, 5, 5, 7, 7, 8, 11.
Oblong 34 X 35 uses 8 squares of size 4, 7, 9, 9, 11, 15, 16, 19.
Oblong 55 X 56 uses 9 squares of size 5, 9, 12, 12, 14, 19, 23, 24, 32.
Oblong 104 X 105 uses 10 squares of size 7, 12, 16, 19, 26, 28, 33, 44, 45, 60.
From _Peter Kagey_, Dec 13 2016: (Start)
An example of the a(10) = 6 squares that can dissect a 10 X 11 oblong:
  +-------+-----------+
  |       |           |
  |   4   |           |
  |       |     6     |
  +---+---+           |
  | 2 | 2 |           |
  +---+---+-+---------+
  |         |         |
  |    5    |    5    |
  |         |         |
  |         |         |
  +---------+---------+
(End)

Ed Pegg Jr, Table of n, a(n) for n = 1..387
S. Anderson, Catalogues of Simple Perfect Squared Rectangles (SPSR)
B. Felgenhauer, Filling Rectangles with Integer-Sided Squares.
Ed Pegg Jr, Minimally Squared Rectangles.
Ed Pegg Jr on StackExchange, Oblongs into minimal squares, Dec 13 2016.

Cf. A005670, A339548.

Corrected term 351 and extended to n=387 by Ed Pegg Jr, Oct 31 2018

A221840 Number of sets of n squares providing dissections of a square.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 2, 4, 7

Offset: 1

Geoffrey H. Morley, Jan 26 2013

It seems that a(11) = 16 and a(12) = 30. [Vladimir Letsko, Sep 17 2013]

			a(8) = 2 as there are two sets of 8 squares which tile a unit square, namely {1(0.75X0.75), 7(0.25X0.25)} and {1(0.6X0.6), 3(0.4X0.4), 4(0.2X0.2)}.

Mathematical Marathon, Problem 152 (in Russian)

Cf. A221841, A005670.

a(10) corrected by Geoffrey H. Morley, Aug 02 2013

A365236 a(n) is the least number of integer-sided squares that can be packed together with the n squares 1 X 1, 2 X 2, ..., n X n to fill out a rectangle.

Original entry on oeis.org

0, 1, 1, 3, 2, 4, 3, 3, 4

Offset: 1

Tamas Sandor Nagy and Thomas Scheuerle, Sep 25 2023

Warning: several terms are provisional as their intended verification effectively assumed the augmenting squares were not larger than n X n. - Peter Munn, Oct 02 2023
The definition does not exclude squares larger than n X n.
Terms for n < 10 were verified by the use of a program.
a(10) <= 5.

			Compositions of rectangles that satisfy the minimal number of augmenting squares for n. Where more than one minimal composition exists for a given n, the table shows a single example. In the table body, the numbers include both the specific mandatory and augmenting squares. a(n) is the total number of squares in the rectangle minus n.
           | 1^2   2^2   3^2   4^2   5^2   6^2   7^2   8^2   9^2  10^2 | Total
  ----------------------------------------------------------------------------
  a(1) = 0 |  1                                                        |   1
  a(2) = 1 |  2     1                                                  |   3
  a(3) = 1 |  2     1     1                                            |   4
  a(4) = 3 |  2     1     2     2                                      |   7
  a(5) = 2 |  2     1     1     2     1                                |   7
  a(6) = 4 |  2     1     3     2     1     1                          |  10
  a(7) = 3 |  1     1     1     3     1     2     1                    |  10
  a(8) = 3 |  3     2     1     1     1     1     1     1              |  11
  a(9) = 4 |  2     2     2     2     1     1     1     1     1        |  13

Tamas Sandor Nagy, Examples for a(1) - a(4).
Tamas Sandor Nagy, Example for a(5).
Tamas Sandor Nagy, Example for a(6).
Tamas Sandor Nagy, Original upper bound examples for a(7) with 5 augmenting squares and a(8) with 6 augmenting squares.
Tamas Sandor Nagy, Example of a conjectured solution for a(10) with 5 augmenting squares, found by Peter Munn.
Thomas Scheuerle, Example for a(6) with smallest possible area.
Thomas Scheuerle, Example for a(7).
Thomas Scheuerle, Example for a(8).
Thomas Scheuerle, Example for a(9).
Thomas Scheuerle, Original upper bound example for a(10) with 6 augmenting squares.
Thomas Scheuerle, Example for a(11) = 4 this is at the same time also a conjectured solution for a(10) = 5.

Cf. A000290, A005670, A005842, A038666, A081287.

a(n) <= 1 + Sum_{k = 1 .. ceiling((n - 1)/2)} (n + (1 - k)*floor(n/k) - 2). This upper bound corresponds to placing the squares with length n up to n - floor((n - 1)/2) all in one row. The remaining mandatory squares will then fit naturally into the rectangle n X (1/2)*(2*n - ceiling((n - 1)/2))*(ceiling((n - 1)/2) + 1).

a(n) > a(n - 1) - 2.

Edited by Peter Munn, Oct 04 2023

A232484 Number of size collections in prime "Mrs. Perkins's Quilt" dissections of integer-sided squares into n squares.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 2, 4, 7, 18, 40, 119, 323, 1100, 3594, 13068, 47444

Offset: 1

N. J. A. Sloane, Nov 29 2013

Wynn, Ed. Exhaustive generation of `Mrs. Perkins's quilt' square dissections for low orders. Discrete Math. 334 (2014), 38--47. MR3240464

Ed Wynn, Exhaustive generation of 'Mrs Perkins's quilt' square dissections for low orders, arXiv:1308.5420

Cf. A005670.

A340919 Sorted sizes of the 55 squares used in the first known squared square of dimension 4205 X 4205 found by Roland Sprague in 1938.

Original entry on oeis.org

13, 29, 35, 39, 50, 52, 65, 78, 87, 91, 104, 116, 117, 130, 140, 143, 145, 174, 182, 195, 203, 221, 232, 234, 247, 261, 270, 286, 290, 299, 305, 312, 319, 325, 340, 375, 406, 429, 435, 493, 522, 551, 565, 575, 615, 638, 665, 667, 696, 705, 725, 957, 1040, 1885, 2320

Offset: 1

Hugo Pfoertner, Feb 16 2021

Stuart Anderson, Roland Percival Sprague (1894-1967), history and bibliography on squaring.net.
James Grime, A Nice Square, Numberphile video, 2017.
David Moews, Squared rectangles, 2007 (provides Bouwcamp code of square).
Roland P. Sprague, Beispiel einer Zerlegung des Quadrats in lauter verschiedene Quadrate, Mathematische Zeitschrift, 45, 607-608, 1939.
R. Sprague, An example of a dissection of the square into pairwise unequal squares, English translation.

Cf. A005670, A006983, A014530, A018835, A089047.

Sum_{k=1..55} a(k)^2 = 4205^2.

cp's OEIS Frontend

A128909 3D version of A005670. The problem is to dissect an n X n X n cube into smaller integer cubes, the gcd of whose sides is 1, using the smallest number of cubes. The gcd condition exclude dissecting a 6 X 6 X 6 cube into eight 3 X 3 X 3 cubes.

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A089047 Edge length of largest square dissectable into up to n squares in Mrs. Perkins's quilt problem.

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A342558 a(n) is the maximum number of distinct currents > 0 in a network of n one-ohm resistors with a total resistance of 1 ohm.

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A089046 Least edge-length of a square dissectable into at least n squares in the Mrs. Perkins's quilt problem.

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A160911 a(n) is the number of arrangements of n square tiles with coprime sides in a rectangular frame, counting reflected, rotated or rearranged tilings only once.

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A279317 Minimal number of squares in a dissection of an (n) X (n+1) oblong into squares.

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A221840 Number of sets of n squares providing dissections of a square.

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A365236 a(n) is the least number of integer-sided squares that can be packed together with the n squares 1 X 1, 2 X 2, ..., n X n to fill out a rectangle.

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A232484 Number of size collections in prime "Mrs. Perkins's Quilt" dissections of integer-sided squares into n squares.

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