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%I #12 Mar 20 2024 09:40:17
%S 1,2,3,6,9,11,18,19,22,27,33,43,51,57,67,99,102,123,163,177,187,267,
%T 627
%N A finite set of numbers relevant for the representation of numbers as primitive distinct sums of three squares (0 square allowed).
%C This set of 23 numbers, possibly with one more number a >= 5*10^10, appears in a corollary of the Halter-Koch reference (Korollar 1.(c), p. 13 with the first line of r_3(n) on p. 11). A number is representable as a^2 + b^2 + c^2 with a,b, and c integers, 0 <= a < b < c, and gcd(a,b,c) = 1 if and only if n is not congruent 0, 4, 7 (mod 8) and not one of the numbers {a{k), k = 1 .. 23}, and, if it exists at all, a further number >= 5*10^10.
%C For the multiplicities of these representable numbers see A224447, and for the numbers themselves see A224448.
%C For a similar set of numbers relevant for sums of three nonzero squares see A051952.
%H F. Halter-Koch, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa42/aa4212.pdf">Darstellung natürlicher Zahlen als Summe von Quadraten</a>, Acta Arith. 42 (1982) 11-20, pp. 13 and 11.
%t representableQ[n_] := Length[ Select[ PowersRepresentations[n, 3, 2], Unequal @@ # && GCD @@ # == 1 & ]] > 0; Select[ Range[1000], Not[ representableQ[#] || MatchQ[ Mod[#, 8], 0 | 4 | 7]] &] (* _Jean-François Alcover_, Apr 10 2013 *)
%Y Cf. A224447, A224448, A051952.
%K nonn,fini
%O 1,2
%A _Wolfdieter Lang_, Apr 09 2013
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