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Search: id:A083357
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| A083357 |
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Numbers n such that A083356(n) (the total area of all incongruent integer-sided rectangles of area <= n) is a square. |
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+0
3
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| 0, 1, 43, 169, 227, 735, 10664, 14702, 78159, 5431210
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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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.
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REFERENCES
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John C. Cock, Solution to Problem 10883 proposed by Nick MacKinnon, Amer. Math. Monthly, 110 (2003), pp. 343-344.
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EXAMPLE
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A083356(43)=2116=46^2, so 43 is in this sequence.
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MATHEMATICA
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For[n=area=0, True, n++; area+=n*Ceiling[DivisorSigma[0, n]/2], If[IntegerQ[s=Sqrt[area]], Print[{n, s}]]]
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CROSSREFS
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Cf. A083356, A083358.
Sequence in context: A123040 A142016 A140640 this_sequence A158604 A057816 A162295
Adjacent sequences: A083354 A083355 A083356 this_sequence A083358 A083359 A083360
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KEYWORD
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nonn,more
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AUTHOR
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Dean Hickerson (dean.hickerson(AT)yahoo.com), Apr 26 2003
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EXTENSIONS
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There are no more terms up to 2*10^7.
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