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%I #14 Jul 11 2018 05:31:27
%S 0,1,43,169,227,735,10664,14702,78159,5431210,8350707565
%N Numbers n such that A083356(n) (the total area of all incongruent integer-sided rectangles of area <= n) is a square.
%C 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?
%C A heuristic argument suggests that the sequence is infinite and has about 2*sqrt(log(n)) terms <= n.
%C No other terms below 10^10.
%H Nick MacKinnon, <a href="https://www.jstor.org/stable/2695719">Problem 10883</a>, Amer. Math. Monthly, 108 (2001) 565; <a href="https://www.jstor.org/stable/3647894">solution</a> by John C. Cock, 110 (2003) 343-344.
%e A083356(43)=2116=46^2, so 43 is in this sequence.
%t For[n=area=0, True, n++; area+=n*Ceiling[DivisorSigma[0, n]/2], If[IntegerQ[s=Sqrt[area]], Print[{n, s}]]]
%Y Cf. A083356, A083358.
%K nonn,more
%O 1,3
%A _Dean Hickerson_, Apr 26 2003
%E a(11) from _Max Alekseyev_, Jan 30 2012
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