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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. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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

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Last modified September 30 14:57 EDT 2020. Contains 337439 sequences. (Running on oeis4.)