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A083357
Numbers n such that A083356(n) (the total area of all incongruent integer-sided rectangles of area <= n) is a square.
2
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
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
Sequence in context: A123040 A142016 A140640 * A158604 A057816 A162295
KEYWORD
nonn,more
AUTHOR
Dean Hickerson, Apr 26 2003
EXTENSIONS
a(11) from Max Alekseyev, Jan 30 2012
STATUS
approved