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

1,3

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?

Links

Table of n, a(n) for n=1..42.

Example

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.

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 *)

Crossrefs

Cf. A336205.

Sequence in context: A090445 A228346 A269996 * A336458 A348867 A018318

Adjacent sequences: A336237 A336238 A336239 * A336241 A336242 A336243

Keyword

nonn

Author

Robert Israel, Jul 13 2020

Extensions

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

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

Status

approved