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%I #27 Jan 01 2016 16:47:31
%S 1,2,3,4,5,6,7,8,9,10,11,12,13,15,16,17,18,19,20,22,23,24,25,27,28,31,
%T 32,33,34,36,37,40,43,44,47,48,52,55,58,60,64,67,68,72,73,76,80,82,88,
%U 92,96,97,100,103,108,112
%N Numbers that are not the sum of 4 distinct squares.
%C It follows from the formula that there are infinitely many integers that cannot be partitioned into a sum of four distinct squares of nonnegative integers and infinitely many that can. Furthermore, the largest odd number that has no such partition is 103, and thereafter the terms satisfy the thirty-first order recurrence relation a(n) = 4a(n-31). - _Ant King_, Nov 02 2010
%H Gordon Pall, <a href="http://www.jstor.org/stable/2301257">On Sums of Squares</a>, The American Mathematical Monthly, Vol. 40, No. 1, (January 1933), pp. 10-18. [From _Ant King_, Nov 02 2010]
%H <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F Let k>=0. Then the only integers that cannot be partitioned into a sum of four distinct squares of nonnegative integers are 4^k * N3, where N3 = (N1 union N2), and N1 and N2 are defined by N1 = {1,3,5,7,9,11,13,15,17,19,23,25,27,31,33,37,43,47,55,67,73,97,103} and N2 = {2,6,10,18,22,34,58,82}, respectively. - _Ant King_, Nov 02 2010
%t data = Reduce[ w^2 + x^2 + y^2 + z^2 == # && 0 <= w < x < y < z < #, {w, x, y, z}, Integers] & /@ Range[112]; DeleteCases[ Table[If[Head[data[[k]]] === Symbol, k, 0], {k, 1, Length[data]}], 0] (* _Ant King_, Nov 02 2010 *)
%Y Cf. A001944 (complement).
%K nonn
%O 1,2
%A _N. J. A. Sloane_
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