A096546 Values w associated with A096545(n), sorted on z, then on y and finally on x.

6, 9, 20, 19, 28, 25, 29, 41, 46, 46, 41, 44, 53, 58, 54, 67, 70, 69, 85, 72, 75, 90, 82, 71, 76, 81, 84, 87, 87, 87, 97, 88, 93, 88, 89, 90, 108, 96, 105, 110, 113, 116, 134, 139, 122, 103, 121, 108, 126, 111, 115, 120, 123, 127, 141, 132, 129, 160, 137, 159, 145, 171

Offset: 1

Author

Lekraj Beedassy, Jun 25 2004

Keywords

• nonn

Comments

For 0

Examples

			Entry 87, for example, is associated with primitive quadruples (x, y, z, w)= (26, 55, 78, 87), (38, 48, 79, 87), (20, 54, 79, 87) satisfying x^3 + y^3 + z^3 = w^3, for 0<x<y<z=A096545(n), with n=28, 29, 30.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..13798 (terms corresponding to z <= 8000)
• Ajai Choudhry, On equal sums of cubes, Rocky Mountains Journal of Mathematics, 28 (4), 1998.
• Fred Richman, Sums of Three Cubes

Crossrefs

Primitive quadruples (x, y, z, w) = (A095868, A095867, A096545, A096546).

Programs

Mathematica

s[w_] := Solve[0 < x < y < z && x^3 + y^3 + z^3 == w^3 && GCD[x, y, z, w] == 1, {x, y, z}, Integers];
xyzw = Reap[For[w = 1, w <= 200, w++, sw = s[w]; If[sw != {}, Print[{x, y, z, w} /. sw; Sow[{x, y, z, w} /. sw ]]]]][[2, 1]] // Flatten[#, 1]&;
SortBy[xyzw, {#[[3]]&, #[[2]]&, #[[1]]&}][[All, 4]] (* Jean-François Alcover, Mar 06 2020 *)

Formula

From Thomas Scheuerle, Jan 29 2025: (Start)

Ajai Choudhry gave this beautiful solution for this problem:

v1,v2,v3 are integers > 0, v2 > v1, v3 > floor((v1^3+v2^3)^(1/3)).

x*d = v3*( -v2^3 - v1^3 + v3^3).

y*d = -v2^4 + 2*v2^3*v1 - 3*v2^2*v1^2 + 2*v2*v1^3 - v1^4 + (v1+v2)*v3^3.

z*d = v2^4 - 2*v2^3*v1 + 3*v2^2*v1^2 - 2*v2*v1^3 + v1^4 + (2*v2-v1)*v3^3.

w*d = v3*( v2^3 + (v2-v1)^3 + v3^3).

d = gcd(x*r, y*r, z*r, w*r). (End)

Extensions

Extended by Ray Chandler, Jun 28 2004

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