A336240 Numbers k such that k = x^2+y^2+z^2 = x^3+y^3+z^3 for some integers x,y,z.

0, 1, 2, 3, 6, 27, 29, 354, 729, 2027, 6859, 7778, 19846, 20577, 23277, 35937, 58754, 130979, 132651, 232282, 265602, 332750, 389017, 499853, 885602, 970299, 1492779, 2146689, 2413154, 3764477, 4330747, 5694978, 5929741, 8120601, 8388227, 12068354, 14348907, 17005629, 23522402, 24137569, 31999403, 34328125

Offset: 1

Author

Robert Israel, Jul 13 2020

Keywords

• nonn

Comments

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.

Are there other infinite parametric families of solutions?

Examples

			a(6)=27 is in the sequence because 27 = (-3)^2 + 3^2 + 3^2 = (-3)^3 + 3^3 + 3^3.
a(7)=29 is in the sequence because 29 = (-2)^2 + (-3)^2 + 4^2 = (-2)^3 + (-3)^3 + 4^3.
		

Crossrefs

Cf. A336205.

Programs

Maple

N:= 2*10^5: # for all terms <= N
R:= NULL:
for xx from 0 while 3*xx^2 <= N do
  for yy from xx while xx^2 + 2*yy^2 <= N do
    for zz from yy while xx^2 + yy^2 + zz^2 <= N do
      t:= xx^2 + yy^2 + zz^2;
      c:= [xx^3,yy^3,zz^3];
      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
od od od:
sort(convert({R},list));

Mathematica

NN = 2*10^5; (* for all terms <= NN *)
R = {};
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]]]]]];
R // Union (* Jean-François Alcover, Aug 11 2023, after Robert Israel *)

Extensions

a(27)-a(35) from David A. Corneth, Jul 13 2020

a(36)-a(42) from Andrew R. Booker, Jul 14 2020