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%I #25 Nov 06 2020 10:38:25
%S 5,8,11,13,14,16,17,19,21,22,23,24,25,26,27,28,30,31,34,36,39,42,57,60
%N Numbers that are the sum of 5 nonzero squares in exactly 1 way.
%C The sequence is likely to be finite and complete as the next term, if it exists, is > 50000.
%C From a proof by _David A. Corneth_ on Nov 08 2017 in A294736: This sequence is complete, see the von Eitzen Link and Price's computation that the next term must be > 50000. Proof. The link mentions "for positive integer n, if n > 5408 then the number of ways to write n as a sum of 5 squares is at least Floor(Sqrt(n - 101) / 8)". So for n > 5408, there are more than one way to write n as a sum of 5 squares. For n <= 5408, it has been verified if n is in the sequence by inspection. Hence the sequence is complete.
%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.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SquareNumber.html">Square Number.</a>
%H <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A243148(a(n),5) = 1. - _Alois P. Heinz_, Feb 25 2019
%t Select[Range[100], Length[Select[PowersRepresentations[#, 5, 2], #[[1]] > 0&]] == 1&] (* _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, 5], {n, 0, 100}], 1] - 1 // Flatten (* _Jean-François Alcover_, Nov 06 2020, after _Alois P. Heinz_ in A243148 *)
%Y Cf. A025429, A025357, A243148.
%K nonn,fini,full
%O 1,1
%A _Robert Price_, Nov 06 2017
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