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%I #26 Oct 27 2017 21:27:40
%S 33,60,105,90,132,177,145,201,225,180,297,228,213,265,345,258,305,300,
%T 393,369,465,417,385,337,420,425,513,537,409,473,449,585,561,545,481,
%U 505,633,540,528,705,593,462,665,681,617,825,564,753,548,689,777,713
%N Largest number with exactly n representations as sum of five positive squares (or 0 if no number with exactly n representations exists).
%C a(0) = 33 has been asked as a riddle by Gerhard Woeginger in de.rec.denksport.
%C There is no number <= 10^6 that is the sum of five positive squares in exactly 188 ways. - _Donovan Johnson_, Aug 15 2013
%C Donovan Johnson's exhaustive search in fact shows that a(188) is undefined / should be assigned the default value 0. _Hagen von Eitzen_, Jun 05 2014
%C In what sense the search is exhaustive? Is any of the 0's of the b-file proved? - _M. F. Hasler_, Oct 27 2017
%H Hagen von Eitzen, <a href="/A080673/b080673.txt">Table of n, a(n) for n = 0..39054</a> [This replaces an earlier b-file computed by Donovan Johnson]
%H H. v. Eitzen, in reply to user James47, <a href="http://math.stackexchange.com/questions/811824/">What is the largest integer with only one representation as a sum of five nonzero squares?</a>, on stackexchange.com, May 2014
%F a(n) = max { k | A025429(k) = n }. - _M. F. Hasler_, May 30 2014
%e A known result says a(0) = 33, since there is no representation of 33 as sum of 5 positive squares.
%e a(1) = 60 is the largest number allowing exactly one such representation.
%t max = 1000; m = Ceiling[Sqrt[max]]; xx = Array[x, 6, 0]; x[0] = 1; iter = Sequence @@ Thread[{Rest[xx], Most[xx], m}]; representations = Table[ Rest[xx] . Rest[xx], Evaluate[iter]] // Flatten // Sort // Split // Select[#, First[#] <= max &] &; counts = {First[#], Length[#]} & /@ representations; a[0] = Complement[Range[max], counts[[All, 1]]] // Last; a[n_] := Select[counts, #[[2]] == n &] // Last // First; Table[a[n], {n, 0, 51}] (* _Jean-François Alcover_, Jul 12 2012 *)
%Y Cf. A080654, A025429.
%K easy,nice,nonn
%O 0,1
%A _Rainer Rosenthal_, Mar 02 2003
%E a(15)-a(51) from _Donovan Johnson_, Aug 23 2010
%E Definition adjusted to cope with otherwise undefined values and b-file extended by _Hagen von Eitzen_, Jun 04 2014