OFFSET
1,1
COMMENTS
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).
LINKS
Alon Amit, How do you find the positive integer solutions to ...?, Quora, Aug 07, 2017.
Andrew Bremner and Allan Macleod, An Unusual Cubic Representation Problem, Annales Mathematicae et Informaticae, volume 43 (2014), pages 29-41.
MathStackExchange, Find integer in the form: a/(b+c) + b/(c+a) + c/(a+b).
H. Nakao, Rational Points on Elliptic Curves: x/(y+z)+y/(z+x)+z/(x+y)=n, 2018 (in Japanese).
EXAMPLE
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).
PROG
(Magma)
is_A369896 := function(k)
E := EllipticCurve([0, 4*k^2 + 12*k - 3, 0, 32*(k+3), 0]);
return (Min([g[1] : g in Generators(E)]) lt 0);
end function;
[k : k in [4..200] | is_A369896(k)];
(Sage)
def is_A369896(k):
E = EllipticCurve([0, 4*k^2 + 12*k - 3, 0, 32*(k+3), 0])
return ((E.rank() > 0) and (min([g.xy()[0] for g in E.gens()]) < 0))
print([k for k in range(1, 70) if is_A369896(k)])
CROSSREFS
KEYWORD
nonn
AUTHOR
Robin Visser, Feb 04 2024
STATUS
approved