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

1,1

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.

References

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

Links

Table of n, a(n) for n=1..49.

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

Formula

a(n) = A336328(n, 3)

Example

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.

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).

Sequence in context: A152308 A072052 A333391 * A034848 A168110 A139972

Adjacent sequences: A336329 A336330 A336331 * A336333 A336334 A336335

Keyword

nonn

Author

Bernard Schott, Jul 20 2020

Status

approved