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A129947 Smallest possible side length for simple perfect squared square of order n; or 0 if no such square exists. 1
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 (list; graph; refs; listen; history; internal format)
OFFSET

1,21

COMMENTS

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) ...

Upper bounds for terms a(30) to a(78);

Most terms > a(30) taken from Ian Gambini's 1999 thesis:

a(30) <= 237, a(31) <= 236,

a(32) <= 414, a(33) <= 499, a(34) <= 389, a(35) <= 480, a(36) <= 448, a(37) <= 476,

a(38) <= 550, a(39) <= 360, a(40) <= 510, a(41) <= 557, a(42) <= 501, a(43) <= 604,

a(44) <= 623, a(45) <= 543, a(46) <= 601, a(47) <= 691, a(48) <= 621, a(49) <= 779,

a(50) <= 788, a(51) <= 853, a(52) <= 976, a(53) <= 824, a(54) <= 971, a(55) <= 939,

a(56) <= 929, a(57) <= 985, a(58) <=1100, a(59) <=1060, a(60) <=1097, a(61) <=1043,

a(62) <=1115, a(63) <=1171, a(64) <=1263, a(65) <=1365, a(66) <=1174, a(67) <=1335,

a(68) <=1394, a(69) <=1410, a(70)<=29448, a(71) <=1443, a(72)<=35051, a(73)<=28412,

a(74)<=30713, a(75) <=1412, a(76)<=39914, a(77)<=32563, a(78)<=26728 ...

REFERENCES

Skinner, J. D. II. Squared Squares: Who's Who & What's What. Published by the author, 1993.

Gambini, Ian. Thesis; 'Quant aux carrés carrelés' L’Universite de la Mediterranee

Aix-Marseille II 1999  pp73-78

LINKS

Anderson, S., Perfect Squared Rectangles and Squared Squares.

Eric Weisstein, Link to a section of The World of Mathematics. Perfect Square Dissection.

CROSSREFS

Cf. A006983 = number of simple perfect squared squares of order n.

Cf. A174386 = smallest possible area for a perfect isosceles right triangled square of order n.

Sequence in context: A103849 A010032 A190026 * A096680 A109383 A036301

Adjacent sequences:  A129944 A129945 A129946 * A129948 A129949 A129950

KEYWORD

more,nonn

AUTHOR

Alexander Adamchuk (alex(AT)kolmogorov.com), Jun 09 2007, corrected Jun 11 2007

EXTENSIONS

Unproved statement misattributed to Skinner replaced, known upper bounds corrected, and crossref added by Geoffrey H. Morley (ghmorley(AT)googlemail.com), Mar 19 2010

Additional term added, initial term a(0)=1 deleted by Stuart E Anderson (stuart.errol.anderson(AT)gmail.com), Dec 26, 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(29)<=221, from Stuart E Anderson & Ed Pegg Jr, Jan 20, 2011

a(29) confirmed as 221, from Stuart E Anderson & Ed Pegg Jr & Stephen Johnson, Aug 22 2011.

New bound for a(31)<=236, computed by Stephen Johnson in September 2011, updated by Stuart E Anderson, Oct 04 2011

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Last modified February 15 14:33 EST 2012. Contains 205818 sequences.