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%I #16 Jun 21 2013 06:00:49
%S 33,38,41,51,54,57,59,62,69,74,77,81,83,90,94,98,99,102,105,107,113,
%T 117,118,121,122,123,125,126,137,138,139,141,150,154,155,158,162,165,
%U 170,177,178,181,187,195,197,203,210,211,213,214,217,218,225,226,229
%N Positive numbers that are the sum of three nonzero squares with no common factor > 1 in exactly two ways.
%C These are the increasingly ordered numbers a(n) for which A223730(a(n)) = 2. See also A223731. These are the numbers n with exactly two 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: a(147) = 1885 = 16^2 + 27^2 + 30^2 = 12^2 + 29^2 + 30^2 is the largest element of this sequence. - _Alois P. Heinz_, Apr 06 2013
%H Alois P. Heinz, <a href="/A223733/b223733.txt">Table of n, a(n) for n = 1..147</a>
%F This sequence lists the increasingly ordered distinct members of the set S2:= {m positive integer | m = a^2 + b^2 + c^2, 0 < a <= b <= c, and there are exactly two different solutions for this m}.
%e a(1) = 33 because the smallest number n with A223730(n) = 2 is 33. The two representations of 33 are denoted by [1, 4, 4], and [2, 2, 5].
%e The two representations for a(n) for n = 2..10 are denoted by
%e n=2, 38: [1, 1, 6], [2, 3, 5],
%e n=3, 41: [1, 2, 6], [3, 4, 4],
%e n=4, 51: [1, 1, 7], [1, 5, 5],
%e n=4, 54: [1, 2, 7], [2, 5, 5], ([3, 3, 6] is non-primitive)
%e n=5, 57: [2, 2, 7], [4, 4, 5],
%e n=6, 59: [1, 3, 7], [3, 5, 5],
%e n=7, 62: [1, 5, 6], [2, 3, 7],
%e n=8, 69: [1, 2, 8], [2, 4, 7],
%e n=9, 74: [1, 3, 8], [3, 4, 7],
%e n=10, 77: [2, 3, 8], [4, 5, 6].
%t threeSquaresCount[n_] := Length[ Select[ PowersRepresentations[n, 3, 2], Times @@ #1 != 0 && GCD @@ #1 == 1 & ]]; Select[ Range[300], threeSquaresCount[#] == 2 &] (* _Jean-François Alcover_, Jun 21 2013 *)
%Y Cf. A223730, A223731, A223732, A223734.
%K nonn
%O 1,1
%A _Wolfdieter Lang_, Apr 05 2013
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