A083357 Numbers n such that A083356(n) (the total area of all incongruent integer-sided rectangles of area <= n) is a square.

0, 1, 43, 169, 227, 735, 10664, 14702, 78159, 5431210, 8350707565

Offset: 1

Author

Dean Hickerson, Apr 26 2003

Keywords

• nonn
• more

Comments

The reference asks "Let R(n) be the set of all rectangles whose side lengths are natural numbers and whose area is at most n. Find an integer n>1 such that the members of R(n), each used exactly once, tile a square.". It shows that n=43 is the smallest solution. A necessary condition is that n be in this sequence. Is this also a sufficient condition?

A heuristic argument suggests that the sequence is infinite and has about 2*sqrt(log(n)) terms <= n.

No other terms below 10^10.

Examples

			A083356(43)=2116=46^2, so 43 is in this sequence.
		

Links

• Nick MacKinnon, Problem 10883, Amer. Math. Monthly, 108 (2001) 565; solution by John C. Cock, 110 (2003) 343-344.

Crossrefs

Cf. A083356, A083358.

Programs

Mathematica

For[n=area=0, True, n++; area+=n*Ceiling[DivisorSigma[0, n]/2], If[IntegerQ[s=Sqrt[area]], Print[{n, s}]]]

Extensions

a(11) from Max Alekseyev, Jan 30 2012