A336329 When F is the Fermat point of a triangle ABC, this sequence lists the integer total distances FA + FB + FC corresponding to primitive triangles in A336328.

112, 147, 185, 283, 273, 331, 331, 403, 559, 485, 645, 520, 691, 592, 637, 965, 1047, 1560, 1415, 1688, 1649, 2093, 1895, 2045, 1687, 1843, 2073, 1839, 1768, 1805, 1729, 1729, 2593, 2337, 2792, 2408, 2709, 2696, 2813, 2704, 2960, 3192, 3007, 3681, 3217, 3752, 2855

Offset: 1

Author

Bernard Schott, Jul 18 2020

Keywords

• nonn

Comments

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

The triples of sides (a,b,c) with a < b < c 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.

For the terms of the data, every FA, FB, FC is a fraction but FA + FB + FC is an integer (see example).

This sequence is not increasing. For example, a(5) = 283 for triangle with largest side = 205 while a(6) = 273 for triangle with largest side = 208. Also, a(6) = a(7) = 331 show that two distinct triangles can have the same minimum possible integer distance FA + FB + FC.

Examples

			For first triple (57, 65, 73), d = 112 is solution of 3*(57^4 + 65^4 + 73^4 + d^4) = (57^2 + 65^2 + 73^2 + d^2)^2, hence, 112 is a term because d = FA + FB + FC = 264/7 + 195/7 + 325/7 = 112.
		

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), A336330 (smallest side), A336331 (middle side), A336332 (largest side), A336333 (perimeter), A351477.

Cf. A061281 (supersequence with non-primitive terms).

Programs

PARI

lista(nn) = my(d); for(c=4, nn, for(b=ceil(c/sqrt(3)), c-1, for(a=1+(sqrt(4*c^2-3*b^2)-b)\2, b-1, if(gcd([a, b, c])==1 && issquare(6*(a^2*b^2+b^2*c^2+c^2*a^2)-3*(a^4+b^4+c^4), &d) && issquare((a^2+b^2+c^2+d)/2, &d), print1(d, ", "))))); \\ Jinyuan Wang, Jul 20 2020

Formula

For triangle (a, b, c) whose area is S, and d = FA+FB+FC, then

d = sqrt((1/2)*(a^2+b^2+c^2) + 2*S*sqrt(3)), also,

d = sqrt(((a^2 + b^2 + c^2)/2) + (1/2) * sqrt(6*(a^2*b^2 + b^2*c^2 + c^2*a^2) - 3*(a^4 + b^4 + c^4))), or

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

Extensions

More terms from Jinyuan Wang, Jul 20 2020