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%I #25 Jul 27 2021 12:27:18
%S 3,5,6,9,10,13,14,15,16,17,18,19,20,21,26,29,30,31,35,36,38,40,41,44,
%T 47,51,53,54,57,62,63,64,66,67,69,70,71,72,73,74,76,77,83,84,86,87,92,
%U 94,96,98,99,101,102,103,105,106,107,108,109,110,112,113,116
%N Integers expressible as (x^3 + y^3 + z^3)/(x*y*z) with nonzero integers x, y and z. Alternatively, integers expressible as a/b + b/c + c/a with nonzero integers a, b and c.
%C A representation k = (x^3 + y^3 + z^3)/(x*y*z) is equivalent to 2 representations k = a/b + b/c + c/a, given by a=y^2*z, b=z^2*x, c=x^2*y and a=y*z^2, b=x*y^2, c=z*x^2. - _Dean Hickerson_, Jul 14 2003
%C For each a(n) > 5 there are infinitely many representations. - _David J. Rusin_, Jul 15 2003
%H Jinyuan Wang, <a href="/A085705/b085705.txt">Table of n, a(n) for n = 1..1000</a>
%H David J. Rusin, <a href="http://www.math.niu.edu/~rusin/research-math/abcn/">For which values of n is a/b + b/c + c/a = n solvable?</a> [Broken link]
%H David J. Rusin, <a href="/A072716/a072716.pdf">For which values of n is a/b + b/c + c/a = n solvable?</a> [Cached copy of html wrapper for paper but in pdf format (so none of the links work)]
%H David J. Rusin, <a href="/A072716/a072716.txt">For which values of n is a/b + b/c + c/a = n solvable?</a> [Cached copy of .txt file]
%e 15 is in the sequence because 15 = (7^3 + (-3)^3 + (-1)^3)/(7*-3*-1) = (7^3 - 3^3 - 1^3)/(7*3*1) = (343 - 27 - 1)/21. This is equivalent to 15 = -9/7 - 7/147 + 147/9 or 15 = -3/63 - 63/49 + 49/3.
%e 16 = (70^3 + (-31)^3 + (-9)^3)/(70*-31*-9) = (70^3 - 31^3 - 9^3)/(70*31*9) = (343000 - 29791 - 729)/19530.
%o (PARI) is(k) = abs(k-4)==1 || ellanalyticrank(ellinit([0, k^2, 0, -72*k, -16*(4*k^3+27)]))[1]; \\ _Jinyuan Wang_, Jul 27 2021
%Y Cf. A072716 (representation by positive x, y, z).
%K nonn
%O 1,1
%A _Hugo Pfoertner_, Jul 18 2003
%E More terms from _David J. Rusin_, Jul 26 2003
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