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

Author

Bernard Schott, Feb 12 2022

Keywords

• nonn

Comments

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

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

Examples

			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.
		

Links

• 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.

Crossrefs

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

Cf. A351476.

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).

Extensions

More terms from Jinyuan Wang, Feb 17 2022