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%I #8 Feb 05 2024 11:00:58
%S 4,6,10,12,14,16,18,24,28,32,34,38,42,46,48,58,60,66,76,82,92,94,98,
%T 102,112,114,116,126,130,132,136,144,146,152,156,158,160,162,166,178,
%U 182,184,186,196,198,200,206,214,218,228,232,244,258,266,268,270,276,282,300,304,310,312,314
%N Positive integers k such that k = a/(b+c) + b/(a+c) + c/(a+b) for some positive integers a, b and c.
%C Bremner and Macleod showed that a positive integer k is in this sequence if and only if the elliptic curve E/Q : y^2 = x^3 + (4*k^2 + 12*k - 3)*x^2 + 32*(k + 3)*x has a generator on the bounded real component of E(R).
%H Alon Amit, <a href="https://www.quora.com/How-do-you-find-the-positive-integer-solutions-to-frac-x-y%2Bz-%2B-frac-y-z%2Bx-%2B-frac-z-x%2By-4/answer/Alon-Amit">How do you find the positive integer solutions to ...?</a>, Quora, Aug 07, 2017.
%H Andrew Bremner and Allan Macleod, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AMI_43_from29to41.pdf">An Unusual Cubic Representation Problem</a>, Annales Mathematicae et Informaticae, volume 43 (2014), pages 29-41.
%H MathStackExchange, <a href="https://math.stackexchange.com/questions/402537/find-integer-in-the-form-fracabc-fracbca-fraccab">Find integer in the form: a/(b+c) + b/(c+a) + c/(a+b)</a>.
%H H. Nakao, <a href="http://www.kaynet.or.jp/~kay/misc/de95.html">Rational Points on Elliptic Curves: x/(y+z)+y/(z+x)+z/(x+y)=n</a>, 2018 (in Japanese).
%e There are no positive integer solutions to a/(b+c) + b/(a+c) + c/(a+b) = k for k = 1, 2, or 3. The smallest positive integer solution to a/(b+c) + b/(a+c) + c/(a+b) = 4 is (a, b, c) = (4373612677928697257861252602371390152816537558161613618621437993378423467772036, 36875131794129999827197811565225474825492979968971970996283137471637224634055579, 154476802108746166441951315019919837485664325669565431700026634898253202035277999).
%o (Magma)
%o is_A369896 := function(k)
%o E := EllipticCurve([0, 4*k^2 + 12*k - 3, 0, 32*(k+3), 0]);
%o return (Min([g[1] : g in Generators(E)]) lt 0);
%o end function;
%o [k : k in [4..200] | is_A369896(k)];
%o (Sage)
%o def is_A369896(k):
%o E = EllipticCurve([0, 4*k^2 + 12*k - 3, 0, 32*(k+3), 0])
%o return ((E.rank() > 0) and (min([g.xy()[0] for g in E.gens()]) < 0))
%o print([k for k in range(1, 70) if is_A369896(k)])
%Y Cf. A283564 (Rank 1).
%K nonn
%O 1,1
%A _Robin Visser_, Feb 04 2024
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