A369896 Positive integers k such that k = a/(b+c) + b/(a+c) + c/(a+b) for some positive integers a, b and c.

4, 6, 10, 12, 14, 16, 18, 24, 28, 32, 34, 38, 42, 46, 48, 58, 60, 66, 76, 82, 92, 94, 98, 102, 112, 114, 116, 126, 130, 132, 136, 144, 146, 152, 156, 158, 160, 162, 166, 178, 182, 184, 186, 196, 198, 200, 206, 214, 218, 228, 232, 244, 258, 266, 268, 270, 276, 282, 300, 304, 310, 312, 314

Offset: 1

Author

Robin Visser, Feb 04 2024

Keywords

• nonn

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).

Examples

			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).
		

Links

• Xianwen Wang, Table of n, a(n) for n = 1..149.
• 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).

Crossrefs

Cf. A283564 (Rank 1).

Programs

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)])