

A180336


Positive numbers l of the form l = A007913(4*k*m^3  k^4), where 1 <= k <= 5*l, 1 <= m <= 5*l.


0



2, 3, 5, 6, 11, 15, 23, 31, 35, 47, 51, 59, 86, 106, 107, 109, 110, 129, 143, 155, 167, 174, 202, 203, 215, 230, 246, 255, 283, 307, 314, 318, 327, 341, 358, 362, 383, 419, 426, 430, 431, 433, 439, 449, 451, 499, 503, 509, 526, 527, 533, 557, 602, 606, 635, 643
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OFFSET

1,1


COMMENTS

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).


LINKS

Table of n, a(n) for n=1..56.


FORMULA

Let a(n)=A007913(4*k*m^3k^4). Put h=sqrt(A008833(4*k*m^3k^4)). Then identity A^3+B^3+C^3=0 is valid for A=2*m^6k^3*m^3k^6+k*(k^3+5*m^3)*h*sqrt(a(n)); B=3*m*(k^3m^3)*(k^2h*sqrt(a(n))); C=k^68*k^3*m^32*m^6k*(k^34*m^3)*h*sqrt(a(n)).


EXAMPLE

We have 2=A007913(4*k*m^3k^4) for k=2,m=3. Therefore a(1)=2; furthermore, 3=A007913(4*k*m^3k^4) for k=m=1. Therefore a(2)=3.


MATHEMATICA

squareFreePart[n_] := Times @@ (#[[1]] ^ Mod[ #[[2]], 2] & /@ FactorInteger@n);
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];
k = 1; lst = {}; While[k < 300, If[ fQ@k, AppendTo[lst, k]]; k++ ]; lst
(* Robert G. Wilson v, Aug 29 2010 *)


CROSSREFS

Cf. A180323, A007913, A008833, A000215.
Sequence in context: A038196 A039849 A039896 * A034407 A068441 A112598
Adjacent sequences: A180333 A180334 A180335 * A180337 A180338 A180339


KEYWORD

nonn,uned


AUTHOR

Vladimir Shevelev, Aug 28 2010, Aug 29 2010


EXTENSIONS

Missing term 35 added and a(14)  a(30) from Robert G. Wilson v, Aug 29 2010
a(31)  a(63) from Robert G. Wilson v, Sep 04 2010


STATUS

approved



