A336330 Smallest side of primitive integer-sided triangles with A < B < C < 2*Pi/3 and such that FA + FB + FC is an integer where F is the Fermat point of the triangle.

57, 73, 43, 127, 97, 111, 49, 95, 296, 152, 323, 147, 285, 255, 247, 469, 403, 871, 561, 657, 559, 1083, 833, 1057, 485, 507, 1072, 760, 767, 379, 211, 195, 1208, 952, 1443, 1023, 1051, 889, 1240, 1209, 1249, 1423, 1005, 1679, 1568, 1843, 193, 485, 1512

Offset: 1

Author

Bernard Schott, Jul 21 2020

Keywords

• nonn

Comments

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

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

If FA + FB + FC = d, then we have this "beautifully symmetric equation" between a, b, c and d (see Martin Gardner):

3*(a^4 + b^4 + c^4 + d^4) = (a^2 + b^2 + c^2 + d^2)^2.

This sequence is not increasing. For example, a(2) = 73 for triangle with largest side = 95 while a(3) = 43 for triangle with largest side = 152.

Examples

			a(1) = 57 because the first triple is (57, 65, 73) with corresponding d = FA + FB + FC = 264/7 + 195/7 + 325/7 = 112 and the symmetric relation satisfies: 3*(57^4 + 65^4 + 73^4 + 112^4) = (57^2 + 65^2 + 73^2 + 112^2)^2 = 642470409.
		

References

• Martin Gardner, Mathematical Circus, Elegant triangles, First Vintage Books Edition, 1979, p. 65.

Links

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

Crossrefs

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

Cf. A072054 (smallest sides: primitives and multiples).

Formula

a(n) = A336328(n, 1).