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%I #20 Jan 03 2016 10:42:52
%S 2,3,5,6,11,15,23,31,35,47,51,59,86,106,107,109,110,129,143,155,167,
%T 174,202,203,215,230,246,255,283,307,314,318,327,341,358,362,383,419,
%U 426,430,431,433,439,449,451,499,503,509,526,527,533,557,602,606,635,643
%N Positive numbers l of the form l = A007913(4*k*m^3 - k^4), where 1 <= k <= 5*l, 1 <= m <= 5*l.
%C The equation x^3 + y^3 + z^3 = 0 is solvable in numbers of the form N + M*sqrt(-a(n)), where M and N are integers. Moreover, it is solvable in numbers of the form N + M*sqrt(-l), where l>0 has the form l = A007913(4*k*m^3 - k^4), where k,m >= 1 (without restrictions k,m <= 5*l). But in this more general case there could be unknown numbers l having this form; this circumstance does not allow construction of the full sequence of such l. Therefore we restrict ourselves by the condition k,m <= 5*l. Note that testing l with respect to this condition is rather simple by sorting all values of k,m <= l. One can prove that, at least, in case the Fermat numbers (A000215) are squarefree, the sequence is infinite. Conjecture (necessity of the form of l): If the equation x^3 + y^3 + z^3 = 0 is solvable in numbers of the form N + M*sqrt(-l) with integer N,M, then there exist positive integers k,m such that l = A007913(4*k*m^3 - k^4).
%F Let a(n)=A007913(4*k*m^3-k^4). Put h=sqrt(A008833(4*k*m^3-k^4)). Then identity A^3+B^3+C^3=0 is valid for A=2*m^6-k^3*m^3-k^6+k*(k^3+5*m^3)*h*sqrt(-a(n)); B=3*m*(k^3-m^3)*(k^2-h*sqrt(-a(n))); C=k^6-8*k^3*m^3-2*m^6-k*(k^3-4*m^3)*h*sqrt(-a(n)).
%e We have 2=A007913(4*k*m^3-k^4) for k=2,m=3. Therefore a(1)=2; furthermore, 3=A007913(4*k*m^3-k^4) for k=m=1. Therefore a(2)=3.
%t squareFreePart[n_] := Times @@ (#[[1]] ^ Mod[ #[[2]], 2] & /@ FactorInteger@n);
%t fQ[n_] := If[b = 0; SquareFreeQ@n, Block[{k = 1, m}, While[k < 5 n + 1, m = -5 n; While[m < 5 n + 1, a = 4 k*m^3 - k^4; If[a > 0, a = squareFreePart@ a, a = 0]; If[a == n, b = a; Print[{a, k, m}]; Goto@ fini, 0]; m++ ]; k++ ]]; Label@ fini; b == n, False];
%t k = 1; lst = {}; While[k < 300, If[ fQ@k, AppendTo[lst, k]]; k++ ]; lst
%t (* _Robert G. Wilson v_, Aug 29 2010 *)
%Y Cf. A180323, A007913, A008833, A000215.
%K nonn,uned
%O 1,1
%A _Vladimir Shevelev_, Aug 28 2010, Aug 29 2010
%E Missing term 35 added and a(14) - a(30) from _Robert G. Wilson v_, Aug 29 2010
%E a(31) - a(63) from _Robert G. Wilson v_, Sep 04 2010
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