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

Author

Rosalie Fay, Oct 16 2017

Keywords

• nonn

Comments

Primitive means that the 4 summands are coprime.

Not every term is the sum of two coprime cubes.

a(1) = A047696(2).

Examples

			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.
		

Links

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

Crossrefs

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

Programs

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 *)