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%I #23 Aug 12 2023 00:56:48
%S 0,1,2,3,6,27,29,354,729,2027,6859,7778,19846,20577,23277,35937,58754,
%T 130979,132651,232282,265602,332750,389017,499853,885602,970299,
%U 1492779,2146689,2413154,3764477,4330747,5694978,5929741,8120601,8388227,12068354,14348907,17005629,23522402,24137569,31999403,34328125
%N Numbers k such that k = x^2+y^2+z^2 = x^3+y^3+z^3 for some integers x,y,z.
%C Contains (2*m^2 + 1)^3 corresponding to x=2*m^2+1, y=-z=2*m^3+m, and m^6/2 - 3*m^2/2 + 3 corresponding to x=-m^2+1, y=-m^3/2+m/2+1, z=m^3/2-m/2+1.
%C Are there other infinite parametric families of solutions?
%e a(6)=27 is in the sequence because 27 = (-3)^2 + 3^2 + 3^2 = (-3)^3 + 3^3 + 3^3.
%e a(7)=29 is in the sequence because 29 = (-2)^2 + (-3)^2 + 4^2 = (-2)^3 + (-3)^3 + 4^3.
%p N:= 2*10^5: # for all terms <= N
%p R:= NULL:
%p for xx from 0 while 3*xx^2 <= N do
%p for yy from xx while xx^2 + 2*yy^2 <= N do
%p for zz from yy while xx^2 + yy^2 + zz^2 <= N do
%p t:= xx^2 + yy^2 + zz^2;
%p c:= [xx^3,yy^3,zz^3];
%p if member(t, {seq(seq(seq(e1*c[1]+e2*c[2]+e3*c[3],e1=[-1,1]),e2=[-1,1]),e3=[-1,1])}) then R:= R, t; fi
%p od od od:
%p sort(convert({R},list));
%t NN = 2*10^5; (* for all terms <= NN *)
%t R = {};
%t Module[{x, y, z, t, c}, For[x = 0, 3*x^2 <= NN, x++, For[y = x, x^2 + 2^2 <= NN, y++, For[z = y, x^2 + y^2 + z^2 <= NN, z++, t = x^2 + y^2 + z^2; c = {x^3, y^3, z^3}; If[MemberQ[Flatten@Table[{e1, e2, e3}. c, {e1, {-1, 1}}, {e2, {-1, 1}}, {e3, {-1, 1}}], t], Print[t]; AppendTo[R, t]]]]]];
%t R // Union (* _Jean-François Alcover_, Aug 11 2023, after _Robert Israel_ *)
%Y Cf. A336205.
%K nonn
%O 1,3
%A _Robert Israel_, Jul 13 2020
%E a(27)-a(35) from _David A. Corneth_, Jul 13 2020
%E a(36)-a(42) from _Andrew R. Booker_, Jul 14 2020
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