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A293647
Positive numbers that are the sum of two (possibly negative) cubes in at least 2 ways (primitive solutions).
4
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
KEYWORD
nonn
AUTHOR
Rosalie Fay, Oct 16 2017
STATUS
approved