A236300 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.

8, 9, 16, 18, 20, 27, 28, 32, 35, 36, 40, 44, 45, 49, 52, 54, 56, 63, 64, 65, 68, 70, 72, 76, 77, 80, 81, 88, 90, 91, 92, 98, 99, 100, 104, 108, 112, 116, 117, 119, 124, 125, 126, 128, 130, 133, 135, 136, 140, 143, 144, 148, 152, 153, 154, 160, 161, 162, 164, 169

Offset: 1

Author

Arkadiusz Wesolowski, Apr 21 2014

Keywords

• nonn

Comments

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.

From Wolfdieter Lang, Apr 30 2014: (Start)

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.

(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.

(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.

(End)

The even numbers qualifying for the case (iib) above are shown in A240227 with the multiplicities A240228. - Wolfdieter Lang, May 02 2014

Examples

			From _Wolfdieter Lang_, Apr 30 2014: (Start)
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, ...
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, ...
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]; ...
(End)
		

Crossrefs

Subsequence of A002808 (the composite numbers). A004432, A025442.