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%I #27 Nov 06 2020 10:38:34
%S 6,9,12,14,15,17,18,20,22,23,25,26,27,28,31,32,34,35,37,40,43
%N Numbers that have exactly one representation as a sum of six positive squares.
%C It appears that this sequence is finite and complete. See the von Eitzen link for a proof for the 5 positive squares case.
%D E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, New York, 1985, p. 86, Theorem 1.
%H H. von 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-nonzer">What is the largest integer with only one representation as a sum of five nonzero squares?</a> on stackexchange.com, May 2014
%H D. H. Lehmer, <a href="http://www.jstor.org/stable/2305380">On the Partition of Numbers into Squares</a>, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.
%F A243148(a(n),6) = 1. - _Alois P. Heinz_, Feb 25 2019
%t m = 6;
%t r[n_] := Reduce[xx = Array[x, m]; 0 <= x[1] && LessEqual @@ xx && AllTrue[xx, Positive] && n == Total[xx^2], xx, Integers];
%t For[n = 0, n < 50, n++, rn = r[n]; If[rn[[0]] === And, Print[n, " ", rn]]] (* _Jean-François Alcover_, Feb 25 2019 *)
%t b[n_, i_, k_, t_] := b[n, i, k, t] = If[n == 0, If[t == 0, 1, 0], If[i<1 || t<1, 0, b[n, i - 1, k, t] + If[i^2 > n, 0, b[n - i^2, i, k, t - 1]]]];
%t T[n_, k_] := b[n, Sqrt[n] // Floor, k, k];
%t Position[Table[T[n, 6], {n, 0, 100}], 1] - 1 // Flatten (* _Jean-François Alcover_, Nov 06 2020, after _Alois P. Heinz_ in A243148 *)
%Y Cf. A000177, A025430, A243148, A294524.
%K nonn,more
%O 1,1
%A _Robert Price_, Nov 25 2017
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