A336332 Largest side, in increasing order, 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.

73, 95, 152, 205, 208, 280, 296, 343, 361, 387, 407, 437, 469, 473, 485, 624, 728, 931, 1016, 1273, 1311, 1313, 1368, 1387, 1443, 1457, 1463, 1469, 1477, 1519, 1560, 1591, 1687, 1895, 2015, 2045, 2045, 2085, 2197, 2231, 2289, 2347, 2363, 2416, 2465, 2553, 2728, 2821, 2923

Offset: 1

Author

Bernard Schott, Jul 20 2020

Keywords

• nonn

Comments

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

This sequence is increasing because triples are in increasing order of largest side.

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.

Examples

			a(36) = a(37) = 2045 is the smallest largest side that appears twice because:
   (1023, 1387, 2045) is a triple with FA+FB+FC = 2408, and
   (1051, 1744, 2045) is a triple with FA+FB+FC = 2709.
		

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), A336330 (smallest side), A336331 (middle side), this sequence (largest side), A336333 (perimeter).

Cf. A072052 (largest sides: primitives and multiples), A333391.

Formula

a(n) = A336328(n, 3).