A217148 Smallest possible side length for a perfect squared square of order n; or 0 if no such square exists.

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, 233, 218, 225, 253, 237

Offset: 1

Author

Geoffrey H. Morley, Sep 27 2012

Keywords

• nonn
• hard
• more

Comments

It is not known whether this sequence is the same as sequence A129947. It may be that A129947(33) = 246 and A217148(33) = 234. - Geoffrey H. Morley, Jan 10 2013

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 upper bounds shown below for 38 and 40-44 are from J. B. Williams. Those for n = 39 and 45-47 are from Gambini's thesis. - Geoffrey H. Morley, Mar 08 2013

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Upper bounds for a(n) for n = 31 to 59

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| +0 +1 +2 +3 +4 +5 +6 +7 +8 +9

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30 | - - - - - - - - 352 360

40 | 328 336 360 413 425 543 601 691 550 583

50 | 644 636 584 685 657 631 751 742 780 958

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The sequence A129947 has identical terms to A217148 (so far), however they are different as A129947 refers to simple perfect squared squares (SPSSs), while A217148 refers to SPSSs and compound perfect squared squares (CPSSs). The simples and compounds together are referred to as perfect squared squares (PSSs). So far it has been observed that all the smallest side lengths belong to SPSSs only. - Stuart E Anderson, Oct 27 2020

Links

• S. E. Anderson, Perfect Squared Rectangles and Squared Squares.
• Stuart E. Anderson, 'Special' Perfect Squared Squares", accessed 2014. - _N. J. A. Sloane_, Mar 30 2014
• I. Gambini, Quant aux carrés carrelés, Thesis, Université de la Méditerranée Aix-Marseille II, 1999, pp. 73-78.
• Ed Pegg Jr., Advances in Squared Squares, Wolfram Community Bulletin, Jul 23 2020
• Eric Weisstein's World of Mathematics, Perfect Square Dissection

Crossrefs

Cf. A129947, A174386, A181735, A217149, A217156.

Extensions

a(29) from Stuart E Anderson added by Geoffrey H. Morley, Nov 23 2012

a(30) from Stuart E Anderson and Lorenz Milla added by Geoffrey H. Morley, Jun 15 2013

a(31) and a(32) from Lorenz Milla and Stuart E Anderson, Oct 05 2013

For additional terms see the Ed Pegg link, also A006983. - N. J. A. Sloane, Jul 29 2020

a(33) to a(37) from J. B. Williams added by Stuart E Anderson, Oct 27 2020