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

%I A096546 #21 Feb 05 2025 22:16:45
%S A096546 6,9,20,19,28,25,29,41,46,46,41,44,53,58,54,67,70,69,85,72,75,90,82,
%T A096546 71,76,81,84,87,87,87,97,88,93,88,89,90,108,96,105,110,113,116,134,
%U A096546 139,122,103,121,108,126,111,115,120,123,127,141,132,129,160,137,159,145,171
%N A096546 Values w associated with A096545(n), sorted on z, then on y and finally on x.
%C A096546 For 0<x<y<z, the primitive quadruples (x,y,z,w) satisfy x^3 + y^3 + z^3 = w^3.
%H A096546 David A. Corneth, <a href="/A096546/b096546.txt">Table of n, a(n) for n = 1..13798</a> (terms corresponding to z <= 8000)
%H A096546 Ajai Choudhry, <a href="https://doi.org/10.1216/rmjm/1181071714">On equal sums of cubes</a>, Rocky Mountains Journal of Mathematics, 28 (4), 1998.
%H A096546 Fred Richman, <a href="http://math.fau.edu/Richman/cubes.htm">Sums of Three Cubes</a>
%F A096546 From _Thomas Scheuerle_, Jan 29 2025: (Start)
%F A096546 Ajai Choudhry gave this beautiful solution for this problem:
%F A096546   v1,v2,v3 are integers > 0, v2 > v1, v3 > floor((v1^3+v2^3)^(1/3)).
%F A096546   x*d =  v3*( -v2^3 - v1^3 + v3^3).
%F A096546   y*d = -v2^4 + 2*v2^3*v1 - 3*v2^2*v1^2 + 2*v2*v1^3 - v1^4 + (v1+v2)*v3^3.
%F A096546   z*d =  v2^4 - 2*v2^3*v1 + 3*v2^2*v1^2 - 2*v2*v1^3 + v1^4 + (2*v2-v1)*v3^3.
%F A096546   w*d =  v3*( v2^3 + (v2-v1)^3 + v3^3).
%F A096546   d = gcd(x*r, y*r, z*r, w*r). (End)
%e A096546 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.
%t A096546 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];
%t A096546 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]&;
%t A096546 SortBy[xyzw, {#[[3]]&, #[[2]]&, #[[1]]&}][[All, 4]] (* _Jean-François Alcover_, Mar 06 2020 *)
%Y A096546 Primitive quadruples (x, y, z, w) = (A095868, A095867, A096545, A096546).
%K A096546 nonn
%O A096546 1,1
%A A096546 _Lekraj Beedassy_, Jun 25 2004
%E A096546 Extended by _Ray Chandler_, Jun 28 2004