A293647 Positive numbers that are the sum of two (possibly negative) cubes in at least 2 ways (primitive solutions).

91, 152, 189, 217, 513, 721, 728, 999, 1027, 1729, 3087, 3367, 4104, 4706, 4921, 4977, 5256, 5859, 6832, 7657, 8587, 8911, 9919, 10621, 10712, 12663, 12691, 12824, 14911, 15093, 15561, 16120, 16263, 20683, 21014, 23058, 23877, 25669, 27937, 28063, 31519, 32984

Offset

1,1

Comments

Primitive means that the 4 summands are coprime.

Not every term is the sum of two coprime cubes.

a(1) = A047696(2).

Links

Robert Israel, Table of n, a(n) for n = 1..1000 (first 352 terms from Rosalie Fay)

Example

189 = 4^3 + 5^3 = 6^3 + (-3)^3 and 4, 5, 6, -3 are coprime, so 189 is in the sequence.
35208 = 34^3 + (-16)^3 = 33^3 + (-9)^3 and 34, -16, 33, -9 are coprime, so 35208 is in the sequence.

Maple

g:= proc(s, n) local x;
  x:= s/2 + sqrt(12*n/s-3*s^2)/6;
  if not x::integer then return NULL fi;
  [x, s - x];
end proc:
filter:= proc(n)
  local pairs, i, j;
  pairs:= map(g, numtheory:-divisors(n), n);
  for i from 2 to nops(pairs) do
    for j from 1 to i-1 do
      if igcd(op(pairs[i]), op(pairs[j]))=1 then return true fi
  od od;
  false
end proc:
select(filter, [seq(seq(9*i+j, j=[1, 2, 7, 8, 9]), i=0..4000)]); # Robert Israel, Oct 22 2017

Mathematica

g[s_, n_] := Module[{x}, x = s/2 + Sqrt[12*n/s - 3*s^2]/6;   If[!IntegerQ[x], Return[Nothing]]; {x, s - x}];
filter[n_] := Module[{pairs, i, j}, pairs = g[#, n]& /@ Divisors[n]; For[i = 2, i <= Length[pairs], i++, For[j = 1, j <= i - 1, j++, If[GCD @@ Join[pairs[[i]], pairs[[j]]] == 1, Return[True]]]]; False];
Select[Flatten[Table[Table[9*i + j, {j, {1, 2, 7, 8, 9}}], {i, 0, 4000}]], filter] (* Jean-François Alcover, May 28 2023, after Robert Israel *)

Crossrefs

Cf. A051347 (all solutions); A018850 (positive cubes); A293648 (only coprime); A293645, A293650

Sequence in context: A088983 A045934 A051347 * A225908 A159961 A113530

Adjacent sequences: A293644 A293645 A293646 * A293648 A293649 A293650

Keyword

nonn

Author

Rosalie Fay, Oct 16 2017

Status

approved