A336205 Numbers k that can be expressed as x^3 + y^3 + z^3 with x^2 + y^2 + z^2 <= k where x, y, z are integers.

0, 1, 2, 3, 6, 7, 8, 9, 10, 15, 16, 17, 18, 19, 20, 24, 25, 26, 27, 28, 29, 34, 35, 36, 37, 38, 43, 45, 46, 48, 53, 54, 55, 56, 57, 60, 61, 62, 63, 64, 65, 66, 69, 71, 72, 73, 80, 81, 83, 88, 90, 91, 92, 97, 98, 99, 100, 101, 106, 109, 116, 117, 118, 119, 120, 123, 124, 125, 126, 127, 128, 129, 132

Offset

1,3

Comments

See A336240 for border case x^2 + y^2 + z^2 = x^3 + y^3 + z^3.

What is the natural density of this sequence?

There are infinitely many infinite parametric families of solutions which have negative values in (x,y,z). For example, 8*(3*a-1)^2*m^6 + 12*(3*a-1)*(a-1)*m^4 - 6*(2*a-1)*m^2 + 2*a^3 + 1 are terms for all a >= 0, m >= 0. (x = 1 - (6*a-2)*m^2, y = a - m*(1-(6*a-2)*m^2), z = a + m*(1-(6*a-2)*m^2)). - Altug Alkan, Jul 17 2020

By definition, corresponding (x,y,z) variables are produced by equation x^3 + y^3 + z^3 = x^2 + y^2 + z^2 + t with t >= 0. That is, x^2*(x-1) + y^2*(y-1) + z^2*(z-1) >= 0. Conjecture: Every even integer can be represented as x^2*(x-1) + y^2*(y-1) + z^2*(z-1) where x, y, z are integers. - Altug Alkan, Jul 19 2020

Links

Robert Israel, Table of n, a(n) for n = 1..8000

Rémy Sigrist, C program for A336205

Example

11 is not a term because there is no (x,y,z) with x^2 + y^2 + z^2 <= 11 when x^3 + y^3 + z^3 = 11.
18 is a term because (-1)^3 + (-2)^3 + 3^3 = 18 and (-1)^2 + (-2)^2 + 3^2 <= 18.
61 is a term because (-4)^3 + 0^3 + 5^3 = 61 and (-4)^2 + 0^2 + 5^2 <= 61.
354 is a term because (-11)^3 + (-8)^3 + 13^3 = (-11)^2 + (-8)^2 + 13^2 = 354.

Maple

filter:= proc(n) local x, y, z, e1, e2;
  for x from 0 while 3*x^2 <= n do
    for y from 0 while x^2 + 2*y^2 <= n do
      for e1 in [-1, 1] do for e2 in [-1, 1] do
        z:= surd(n + e1*x^3 + e2*y^3, 3);
        if z::integer and x^2 + y^2 + z^2 <= n then return true fi;
  od od od od;
  false
end proc:
select(filter, [$0..200]); # Robert Israel, Jul 12 2020

Mathematica

filter[n_] := Module[{x, y, z, e1, e2},
  For[x = 0, 3*x^2 <= n, x++,
    For[y = 0, x^2 + 2*y^2 <= n, y++,
      For[e1 = -1, e1 <= 1, e1 += 2, For[e2 = -1, e2 <= 1, e2 += 2,
        z = (n + e1*x^3 + e2*y^3)^(1/3);
        If[IntegerQ[z] && x^2 + y^2 + z^2 <= n, Return[True]]
  ]]]]; False];
Select[Range[0, 200], filter] (* Jean-François Alcover, Aug 11 2023, after Robert Israel *)

Prog

(C) See Links section.

Crossrefs

Cf. A004825 (subsequence), A060464 (supersequence), A336240.

Sequence in context: A163099 A047287 A039047 * A047246 A039029 A037460

Adjacent sequences: A336202 A336203 A336204 * A336206 A336207 A336208

Keyword

nonn,easy

Author

Altug Alkan, Jul 12 2020

Status

approved