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

1,1

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

Table of n, a(n) for n=1..63.

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

Cf. A283564 (Rank 1).

Sequence in context: A309177 A163164 A137230 * A283564 A348005 A181794

Adjacent sequences: A369893 A369894 A369895 * A369897 A369898 A369899

Keyword

nonn

Author

Robin Visser, Feb 04 2024

Status

approved