A351477 a(n) is the common denominator of FA, FB and FC, where F is the Fermat point of the integer-sided triangle ABC with A < B < C < 2*Pi/3 such that FA + FB + FC = A336329(n).

7, 7, 37, 283, 91, 331, 331, 13, 43, 97, 43, 13, 691, 37, 91, 193, 349, 13, 283, 211, 97, 91, 379, 409, 7, 97, 691, 613, 13, 19, 13, 91, 2593, 19, 349, 43, 1, 337, 97, 169, 37, 19, 31, 409, 3217, 67, 571, 169, 241, 43, 67, 157, 4171, 3601, 889, 1591, 811, 1, 139

Offset

1,1

Comments

Inspired by Project Euler, Problem 143 (see link).

For the corresponding primitive triples, miscellaneous properties and references, see A336328.

Links

Table of n, a(n) for n=1..59.

Leisure Maths Entertainment Forum, The primitive integer triangles with nonzero rational distances between three vertices and 1st isogonic center, Chinese blog.

Project Euler, Problem 143 - Investigating the Torricelli point of a triangle.

Eric Weisstein's World of Mathematics, Fermat points.

Formula

a(n) = A351476(n)/A336329(n).

a(n) is the common denominator of fractions FA, FB, FC when FA = sqrt(((2*b*c)^2 - (b^2+c^2-d^2)^2)/3) / d, FB = sqrt(((2*a*c)^2 - (a^2+c^2-d^2)^2)/3) / d, FC = sqrt(((2*a*b)^2 - (a^2+b^2-d^2)^2)/3) / d, with a = (A336328(n,1), b = (A336328(n,2), c = (A336328(n,3)) and d = A336329(n) (formulas FA, FB, FC from Jinyuan Wang, Feb 17 2022).

Example

For 1st triple (57, 65, 73) in A336328, we get A336329(1) = FA + FB + FC = 325/7 + 264/7 + 195/7 = 112, hence a(1) = 7.
For 3rd triple (43, 147, 152) in A336328, we get A336329(3) = FA + FB + FC = 5016/37 + 1064/37 + 765/37 = 185, hence a(3) = 37.

Crossrefs

Cf. A336328 (primitive triples), A336329 (FA + FB + FC), A336330 (smallest side), A336331 (middle side), A336332 (largest side), A336333 (perimeter).

Cf. A351476.

Sequence in context: A154702 A112685 A201958 * A153721 A286985 A151791

Adjacent sequences: A351474 A351475 A351476 * A351478 A351479 A351480

Keyword

nonn,changed

Author

Bernard Schott, Feb 12 2022

Extensions

More terms from Jinyuan Wang, Feb 17 2022

Status

approved