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%I #41 May 10 2021 21:08:21
%S 5,10,13,14,17,20,21,25,26,29,30,34,35,37,38,40,41,42,45,46,49,50,52,
%T 53,54,56,58,59,61,62,65,66,68,69,70,73,74,75,77,78,80,81,82,83,84,85,
%U 86,89,90,91,93,94,97,98,100,101,104,105,106,107,109,110,113
%N Numbers that are the sum of 3 distinct squares, i.e., numbers of the form x^2 + y^2 + z^2 with 0 <= x < y < z.
%C Also: Numbers which are the sum of two or three distinct nonzero squares. - _M. F. Hasler_, Feb 03 2013
%C According to Halter-Koch (below), a number n is a sum of 3 squares, but not a sum of 3 distinct squares (i.e., is in A001974 but not A000408), if and only if it is of the form 4^j*s, where j >= 0 and s in {1, 2, 3, 6, 9, 11, 18, 19, 22, 27, 33, 43, 51, 57, 67, 99, 102, 123, 163, 177, 187, 267, 627, ?}, where ? denotes at most one unknown number that, if it exists, is > 5*10^10. - _Jeffrey Shallit_, Jan 15 2017
%H T. D. Noe, <a href="/A001974/b001974.txt">Table of n, a(n) for n=1..1000</a>
%H Franz Halter-Koch, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa42/aa4212.pdf">Darstellung natürlicher Zahlen als Summe von Quadraten</a>, Acta Arithmetica 42 (1982), pp. 11-20.
%H <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%e 5 = 0^2 + 1^2 + 2^2.
%t r[n_] := Reduce[0 <= x < y < z && x^2 + y^2 + z^2 == n, {x, y, z}, Integers]; ok[n_] := r[n] =!= False; Select[ Range[113], ok] (* _Jean-François Alcover_, Dec 05 2011 *)
%o (Python)
%o from itertools import combinations
%o def aupto(lim):
%o s = filter(lambda x: x <= lim, (i*i for i in range(int(lim**.5)+2)))
%o s3 = set(filter(lambda x: x<=lim, (sum(c) for c in combinations(s, 3))))
%o return sorted(s3)
%o print(aupto(113)) # _Michael S. Branicky_, May 10 2021
%Y Cf. A001983, A024803, A025339, A025442, A004432.
%Y Cf. A004436 (complement).
%K nonn,easy,nice
%O 1,1
%A _N. J. A. Sloane_
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