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A369896
Positive integers k such that k = a/(b+c) + b/(a+c) + c/(a+b) for some positive integers a, b and c.
1
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
Andrew Bremner and Allan Macleod, An Unusual Cubic Representation Problem, Annales Mathematicae et Informaticae, volume 43 (2014), pages 29-41.
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
KEYWORD
nonn
AUTHOR
Robin Visser, Feb 04 2024
STATUS
approved