

A129947


Smallest possible side length for a simple perfect squared square of order n; or 0 if no such square exists.


12



0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 112, 110, 110, 120, 147, 212, 180, 201, 221, 201, 215, 185, 223, 218, 225, 253, 237
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,21


COMMENTS

A squared rectangle (which may be a square) is a rectangle dissected into a finite number, two or more, of squares. If no two of these squares have the same size the squared rectangle is perfect. The order of a squared rectangle is the number of constituent squares.
A squared rectangle is simple if it does not contain a smaller squared rectangle.
The smallest known sides of simple perfect squared squares (and the known orders of the squares) are 110 (22, 23), 112 (21), 120 (24), 139 (22, 23), 140 (23), 145 (23), 147 (22, 25) ...
The upper bounds shown below for n = 38 and 4044 are from J. B. Williams. The rest are from Gambini's thesis.  Geoffrey H. Morley, Mar 08 2013
======================================
Upper bounds for a(n) for n = 38 to 59
======================================
 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9
======================================================
30          352 360
40  328 336 360 413 425 543 601 691 621 779
50  788 853 ? 824 971 939 929 985 1100 1060
======================================================
(End)


LINKS



CROSSREFS



KEYWORD

nonn,hard,more


AUTHOR



EXTENSIONS

Unproved statement misattributed to Skinner replaced, known upper bounds corrected, and crossref added by Geoffrey H. Morley, Mar 19 2010
Upper bounds for terms a(31) to a(78), (all from Ian Gambini's thesis) added by Stuart E Anderson, Jan 20 2011
New bound for a(31)<=236, computed by Stephen Johnson in September 2011, updated by Stuart E Anderson, Oct 04 2011


STATUS

approved



