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

Author

Altug Alkan, Jul 12 2020

Keywords

• nonn
• easy

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

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..8000
• Rémy Sigrist, C program for A336205

Crossrefs

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

Programs

C

See Links section.

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 *)