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%I #22 May 28 2023 03:14:11
%S 91,152,189,217,513,721,728,999,1027,1729,3087,3367,4104,4706,4921,
%T 4977,5256,5859,6832,7657,8587,8911,9919,10621,10712,12663,12691,
%U 12824,14911,15093,15561,16120,16263,20683,21014,23058,23877,25669,27937,28063,31519,32984
%N Positive numbers that are the sum of two (possibly negative) cubes in at least 2 ways (primitive solutions).
%C Primitive means that the 4 summands are coprime.
%C Not every term is the sum of two coprime cubes.
%C a(1) = A047696(2).
%H Robert Israel, <a href="/A293647/b293647.txt">Table of n, a(n) for n = 1..1000</a> (first 352 terms from Rosalie Fay)
%e 189 = 4^3 + 5^3 = 6^3 + (-3)^3 and 4, 5, 6, -3 are coprime, so 189 is in the sequence.
%e 35208 = 34^3 + (-16)^3 = 33^3 + (-9)^3 and 34, -16, 33, -9 are coprime, so 35208 is in the sequence.
%p g:= proc(s,n) local x;
%p x:= s/2 + sqrt(12*n/s-3*s^2)/6;
%p if not x::integer then return NULL fi;
%p [x,s - x];
%p end proc:
%p filter:= proc(n)
%p local pairs, i,j;
%p pairs:= map(g, numtheory:-divisors(n),n);
%p for i from 2 to nops(pairs) do
%p for j from 1 to i-1 do
%p if igcd(op(pairs[i]),op(pairs[j]))=1 then return true fi
%p od od;
%p false
%p end proc:
%p select(filter, [seq(seq(9*i+j,j=[1,2,7,8,9]),i=0..4000)]); # _Robert Israel_, Oct 22 2017
%t g[s_, n_] := Module[{x}, x = s/2 + Sqrt[12*n/s - 3*s^2]/6; If[!IntegerQ[x], Return[Nothing]]; {x, s - x}];
%t 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];
%t 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_ *)
%Y Cf. A051347 (all solutions); A018850 (positive cubes); A293648 (only coprime); A293645, A293650
%K nonn
%O 1,1
%A _Rosalie Fay_, Oct 16 2017
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