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%I #17 May 06 2014 06:33:57
%S 8,9,16,18,20,27,28,32,35,36,40,44,45,49,52,54,56,63,64,65,68,70,72,
%T 76,77,80,81,88,90,91,92,98,99,100,104,108,112,116,117,119,124,125,
%U 126,128,130,133,135,136,140,143,144,148,152,153,154,160,161,162,164,169
%N Numbers n of the form x^3 + y^3 + z^3 - 3*x*y*z for x,y,z >= 0, where x + y + z < n.
%C x^3 + y^3 + z^3 - 3*x*y*z = (x + y + z)*(x^2 + y^2 + z^2 - x*y - x*z - y*z), hence all terms are composite.
%C From _Wolfdieter Lang_, Apr 30 2014: (Start)
%C Take x >= y >= z >= 0, not all identical: the numbers are of the form (x + y + z)*(u^2 + v^2 + w^2)/2, where u = x-y, v = x-z, w = y-z, with u >= 0, v >=0, w >= 0, u - v + w = 0 and u^2 + v^2 + w^2 >= 4.
%C (i) If, say, x = y but not equal to z, then the numbers are of the form (2*x+y)*(x-z)^2 with x-z >= 2, z >= 0. Similarly for the other case with y = z not equal to x.
%C (ii) If x, y and z are distinct, u >= 1, v >= 1 and w >= 1, hence u is not equal to v, and v is not equal to w (because u - v + w = 0). (iia) If u = w then the numbers are of the form 3*y*3*(y-z)^2 with y-z >= 1, z >= 0. (iib) If the u, v, w are distinct >= 1 then the even members of the sequence A004432 with multiplicities A025442 are of interest. But only those (u, v, w) qualify which satisfy u - v + w = 0. E.g., A025442(5) = 30 = 1^2 + 2^2 + 5^2 does not qualify because no permutation of 1, 2, 5 works for u, v, w. A025442(1) = 14 qualifies because (u, v, w) = (2, 3, 1) satisfies 2 - 3 + 1 = 0. Then [x, y, z] = [4, 2, 1] and the number is 7*14/2 = 49.
%C (End)
%C The even numbers qualifying for the case (iib) above are shown in A240227 with the multiplicities A240228. - _Wolfdieter Lang_, May 02 2014
%e From _Wolfdieter Lang_, Apr 30 2014: (Start)
%e The numbers of type (i) are seq((2*x+z)*(z-x)^2, z=0..(x-2)) (if x >= 2) and seq((2*x+z)*(z-x)^2, z >= (x+2)) for x = 0, 1, 2, ... E.g., x = 3: 54, 28, 44, and 108, 208, 350, 540, 784, 1088, 1458, 1900, 2420, 3024, ...
%e The numbers of type (iia) are [seq(9*y*(y-z)^2, y >= 1+z)] for z = 0, 1, 2, ... E.g., z=3: 36, 180, 486, 1008, 1800, 2916, 4410, ...
%e The numbers of type (iib) come from the even members 14, 26, 30, 38, 42, 46, 50, ... of A025442 (each with multiplicity 1) except of 30 (as explained above in a comment), 46 with 1, 3, 6 which is out, and also 50 with 3, 4, 5. 7*14/2 = 49 (see the comment above); 10*26/2 = 130 from (u, v, w) = (1, 4, 3) and [x, y, z] = [5, 4, 1]; 11*38/2 = 209 from (2, 5, 3) and [6, 4, 1]; 12*42/2 = 252 from (1, 5, 4) and [6, 5, 1]; ...
%e (End)
%Y Subsequence of A002808 (the composite numbers). A004432, A025442.
%K nonn
%O 1,1
%A _Arkadiusz Wesolowski_, Apr 21 2014
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