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

1,3

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.

Links

Table of n, a(n) for n=1..11.

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

Example

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

Mathematica

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

Crossrefs

Cf. A083356, A083358.

Sequence in context: A123040 A142016 A140640 * A158604 A057816 A162295

Adjacent sequences: A083354 A083355 A083356 * A083358 A083359 A083360

Keyword

nonn,more

Author

Dean Hickerson, Apr 26 2003

Extensions

a(11) from Max Alekseyev, Jan 30 2012

Status

approved