A283564 -id:A283564 - cp's OEIS frontend

A371846 The number of digits in max(a,b,c), where a, b, and c are the smallest positive integer solutions to a/(b+c) + b/(a+c) + c/(a+b) = A283564(n).

81, 134, 190, 2707, 1876, 414, 10323, 33644, 81853, 14836, 1584369, 886344, 198771, 418086, 244860, 9188, 215532, 23662, 85465, 252817, 625533, 935970, 112519, 196670, 8572242, 3607937, 26942, 259164, 12046628, 15097279, 1265063, 398605460, 2828781, 20770896, 5442988, 11323026, 726373, 71225279

Offset: 1

Ryan Tang, Apr 09 2024

			For n=1, a/(b+c) + b/(a+c) + c/(a+b) = 4, and the value of a, b, and c are 154476802108746166441951315019919837485664325669565431700026634898253202035277999, 36875131794129999827197811565225474825492979968971970996283137471637224634055579, and 4373612677928697257861252602371390152816537558161613618621437993378423467772036, and the maximum value has 81 digits.

Googology Wiki, Bremner-Macleod numbers.

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

Original entry on oeis.org

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

Robin Visser, Feb 04 2024

nonn

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).

			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).

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).

Cf. A283564 (Rank 1).

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)];

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)])

A366547 Triples of positive integers (a,b,c) with a<=b<=c such that a/(b+c) + b/(c+a) + c/(a+b) = 4.

Original entry on oeis.org

4373612677928697257861252602371390152816537558161613618621437993378423467772036, 36875131794129999827197811565225474825492979968971970996283137471637224634055579, 154476802108746166441951315019919837485664325669565431700026634898253202035277999

Offset: 1

Chai Wah Wu, Oct 12 2023

nonn

Solution of a meme/puzzle that has been circulating on the web. The first 3 terms form the smallest solution.

Andrew Bremner and Allan Macleod, An unusual cubic representation problem, Annales Mathematicae et Informaticae, 43(2014), pp.29-41.
Quora, How do you find the positive integer solutions to x/(y+z)+y/(z+x)+z/(x+y)=4?

Cf. A283564, A309170, A309168, A309177, A309178.

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A371846 The number of digits in max(a,b,c), where a, b, and c are the smallest positive integer solutions to a/(b+c) + b/(a+c) + c/(a+b) = A283564(n).

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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.

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A366547 Triples of positive integers (a,b,c) with a<=b<=c such that a/(b+c) + b/(c+a) + c/(a+b) = 4.

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