A384132 Integers k such that the Diophantine equation x^3 + y^3 + z^3 + w^3 = k^3, where 0 < x < y < z < w has no integer solutions.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 16, 17, 19, 21, 22, 25, 27, 29, 47, 58, 61, 71, 113, 121

Offset

1,2

Comments

Conjecture: a(27)=121 is the largest integer whose cube cannot be described as the sum of four distinct positive cubes.

Links

Table of n, a(n) for n=1..27.

Example

13 is not a term because 13^3 = 5^3 + 7^3 + 9^3 + 10^3 = 1^3 + 5^3 + 7^3 + 12^3.

Maple

filter:= proc(n) local t0, t1, t2, t3, q, x, y, z, w;
  t0:= n^3;
  for x from ceil((t0/4)^(1/3)) do
    t1:= t0 - x^3;
    if t1 < 36 then break fi;
    for y from ceil((t1/3)^(1/3)) do
      t2:= t1 - y^3;
      if t2 < 9 or y >= x then break fi;
      for z from ceil((t2/2)^(1/3)) do
        t3:= t2 - z^3;
        if t3 < 1 or z >= y then break fi;
        w:= iroot(t3, 3, 'q');
        if q and w < z then return false fi;
   od od od;
   true
end proc:
select(filter, [$1..1000]); # Robert Israel, Dec 02 2025

Mathematica

a=Select[Range@125, Length@Select[PowersRepresentations[#^3, 4, 3], 0<#[[1]]<#[[2]]<#[[3]]<#[[4]]&]==0&]

Crossrefs

Cf. A003327, A383877.

Sequence in context: A003964 A023781 A135578 * A050607 A238368 A342164

Adjacent sequences: A384129 A384130 A384131 * A384133 A384134 A384135

Keyword

nonn,more

Author

Zhining Yang, May 20 2025

Status

approved