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A060466 Value of y of the solution to x^3 + y^3 + z^3 = A060464(n) (numbers not 4 or 5 mod 9) with smallest |z| and smallest |y|, 0 <= |x| <= |y| <= |z|.

%I A060466 #41 Apr 06 2020 00:47:17
%S A060466 0,0,1,1,-1,-1,0,1,1,-2,10,2,-1609,2,-2,-2,-2,-14,-15550555555,-1,-1,
%T A060466 0,1,1,-2218888517,-8778405442862239,2,2,2,-3,-3,134476,
%U A060466 80435758145817515,2,-7,-3,3,7,-26,659,60702901317,3,-11,3,-21,-2,-4,-4,3,-1,0,1,1,-4,20,2,9,2
%N A060466 Value of y of the solution to x^3 + y^3 + z^3 = A060464(n) (numbers not 4 or 5 mod 9) with smallest |z| and smallest |y|, 0 <= |x| <= |y| <= |z|.
%C A060466 Indexed by A060464.
%C A060466 Only primitive solutions where gcd(x,y,z) does not divide n are considered.
%C A060466 From the solution A060464(24) = 30 = -283059965^3 - 2218888517^3 + 2220422932^3 (smallest possible magnitudes according to A. Bogomolny), one has a(24) = -2218888517. A solution to A060464(25) = 33 remains to be found. Other values for larger n can be found in the second column of the table on Hisanori Mishima's web page. - _M. F. Hasler_, Nov 10 2015
%D A060466 R. K. Guy, Unsolved Problems in Number Theory, Section D5.
%H A060466 A. Bogomolny, <a href="http://www.cut-the-knot.org/arithmetic/algebra/FinikyDiophantineEquations.shtml">Finicky Diophantine Equations</a> on cut-the-knot.org, accessed Nov. 10, 2015.
%H A060466 A.-S. Elsenhans, J. Jahnel, <a href="http://dx.doi.org/10.1090/S0025-5718-08-02168-6">New sums of three cubes</a>, Math. Comp. 78 (2009) 1227-1230.
%H A060466 K. Koyama, Y. Tsuruoka, H. Sekigawa, <a href="http://dx.doi.org/10.1090/S0025-5718-97-00830-2">On searching for solutions of the Diophantine equation x^3+y^3+z^3=n</a>, Math. Comp. 66 (1997) 841.
%H A060466 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/math04/matb0100.htm">About n=x^3+y^3+z^3</a>
%H A060466 Eric S. Rowland, <a href="https://ericrowland.github.io/papers/Known_families_of_integer_solutions_of_x^3+y^3+z^3=n.pdf">Known families of integer solutions of x^3+y^3+z^3=n</a>
%H A060466 A. Tyszka, <a href="http://arxiv.org/abs/0901.2093">A hypothetical upper bound for the solutions of a Diophantine equation with a finite number of solutions</a>, arXiv:0901.2093 [math.NT], 2009-2014.
%e A060466 For n = 16 the smallest solution is 16 = (-511)^3 + (-1609)^3 + 1626^3, which gives the term -1609.
%e A060466 42 = 12602123297335631^3 + 80435758145817515^3 + (-80538738812075974)^3 was found by Andrew Booker and Andrew Sutherland.
%e A060466 74 = 66229832190556^3 + 283450105697727^3 + (-284650292555885)^3 was found by Sander Huisman.
%t A060466 nmax = 29; A060464 = Select[Range[0, nmax], Mod[#, 9] != 4 && Mod[#, 9] != 5 &]; A060465 = {0, 0, 0, 1, -1, 0, 0, 0, 1, -2, 7, -1, -511, 1, -1, 0, 1, -11, -2901096694, -1, 0, 0, 0, 1}; r[n_, x_] := Reduce[0 <= Abs[x] <= Abs[y] <= Abs[z] && n == x^3 + y^3 + z^3, {y, z}, Integers]; A060466 = Table[y /. ToRules[ Simplify[ r[A060464[[k]], A060465[[k]]] /. C[1] -> 0]], {k, 1, Length[A060464]}] (* _Jean-François Alcover_, Jul 11 2012 *)
%Y A060466 Cf. A060465, A060467, A173515.
%K A060466 sign,nice,hard
%O A060466 0,10
%A A060466 _N. J. A. Sloane_, Apr 10 2001
%E A060466 In order to be consistent with A060465, where only primitive solutions are selected, a(18)=2 was replaced with -15550555555, by _Jean-François Alcover_, Jul 11 2012
%E A060466 Edited and a(24) added by _M. F. Hasler_, Nov 10 2015
%E A060466 a(25) from Tim Browning and further terms added by _Charlie Neder_, Mar 09 2019
%E A060466 More terms from _Jinyuan Wang_, Feb 14 2020