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%I #18 Aug 18 2015 04:42:35
%S 3,6,9,11,14,17,18,19,21,22,26,27,29,30,34,35,42,43,45,46,49,50,53,61,
%T 65,67,70,73,75,78,82,91,93,97,106,109,115,133,142,145,147,157,163,
%U 169,190,193,202,205,235,253,265,277,298,397,403,427,442,445,505,793
%N Positive numbers that are the sum of three nonzero squares with no common factor > 1 in exactly one way.
%C These are the increasingly ordered numbers a(n) for which A233730(a(n)) = 1. See also A233731. These are the numbers n with exactly one representation as a primitive sum of three nonzero squares (not taking into account the order of the three terms, and the number to be squared for each term is taken positive).
%C Conjecture: 793 = 6^2 + 9^2 + 26^2 is the largest element of this sequence. - _Alois P. Heinz_, Apr 06 2013
%H Eugen J. Ionascu, <a href="http://arxiv.org/abs/1508.03643">Ehrhart polynomial for lattice squares, cubes and hypercubes</a>, arXiv:1508.03643 [math.NT], 2015.
%F This sequence lists the increasingly ordered members of the set S1 := {m positive integer | m = a^2 + b^2 + c^2, 0 < a <= b <= c, gcd(a,b,c) = 1, with only one such solution for this m}.
%e a(1) = 3 because there is no solution for m = 1 and 2 as a primitive sum of three nonzero squares, and m = 3 = 1^2 + 1^2 + 1^2 is the only solution with [a,b,c] = [1,1,1].
%e a(5) = 14 because 14 is the fifth largest member of the set S1, and [a,b,c] = [1,2,3] denotes this unique representation for m = 14.
%t threeSquaresCount[n_] := Length[ Select[ PowersRepresentations[n, 3, 2], Times @@ #1 != 0 && GCD @@ #1 == 1 & ]]; Select[ Range[800], threeSquaresCount[#] == 1 &] (* _Jean-François Alcover_, Jun 21 2013 *)
%Y Cf. A233730, A233731, A233733, A233734.
%K nonn
%O 1,1
%A _Wolfdieter Lang_, Apr 05 2013
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