A336166 - OEIS

%I #18 Oct 17 2020 02:11:02

%S 0,1,-3,4,9,-12,16,25,-27,-35,36,37,-48,49,-59,64,-75,81,100,-108,121,

%T 144,-147,-159,169,172,-192,196,225,-227,-243,-255,256,261,-287,289,

%U -300,-311,324,-335,361,-363,373,400,-432,441,484,-507,529,568,576,-588

%N Values z of primitive solutions (x, y, z) to the Diophantine equation x^3 + y^3 + 2*z^3 = 2.

%C Terms are arranged in order of increasing absolute value (if equal, the negative number comes first).

%C Segre shows that 1-(9/2)*A000578(2n), (-3)*A000290(n), and A016754(n) are terms of the sequence.

%D R. K. Guy, Unsolved Problems in Number Theory, D5.

%H Beniamino Segre, <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=46064">On the rational solutions of homogeneous cubic equations in four variables</a>, Math. Notae, 11 (1951), 1-68.

%e (-5)^3 + (-11)^3 + 2 * 9^3 = 2, 9 is a term.

%e (25)^3 + (-23)^3 + 2 * (-12)^3 = 2, -12 is a term.

%t t1 = Union[Plus@@@Tuples[Range[-11643, 11643]^3, 2]];

%t t2 = Table[2 - 2z^3, {z, -588, 588}];

%t t = Select[t1, MemberQ[t2, #] &];

%t u = ((2 - t)/2)^(1/3) /. (-1)^(1/3) -> (-1);

%t v = Table[(-1)^n*Floor[(n + 1)/2], {n, 0, 1176}];

%t Select[v, MemberQ[u, #] &]

%Y Cf. A000290, A000578, A004825, A004826, A016754, A028387, A050791, A130472, A195006.

%K sign

%O 1,3

%A _XU Pingya_, Jul 10 2020