A351801 a(n) = A351477(n) * FA where F is the Fermat point of a primitive integer-sided triangle ABC with A < B < C < 2*Pi/3 and FA + FB + FC = A336329(n).

325, 440, 5016, 39360, 14800, 70720, 91200, 3864, 9405, 30429, 11704, 4669, 250096, 11704, 32640, 81840, 203000, 7208, 218120, 199325, 99360, 76760, 359352, 342912, 8184, 122200, 595595, 621387, 12600, 26040, 19320, 137344, 3108105, 24955, 409640, 58400, 1520

Offset

1,1

Comments

Inspired by Project Euler, Problem 143 (see link) where such a triangle is called a "Torricelli triangle".

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

Equivalently, a(n) is the numerator of the fraction FA = a(n) / A351477(n).

Also, if F is the Fermat point of a triangle ABC with A < B < C < 2*Pi/3, where AB, BC, CA, FA, FB and FC are all positive integers, then, when FA + FB + FC = d = A351476(n), we have FA = a(n).

FA is the largest length with FC < FB < FA (remember a < b < c).

Links

Table of n, a(n) for n=1..37.

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

Wikipedia, Fermat point.

Formula

a(n) = A351476(n) - A351802(n) - A351803(n).

FA = sqrt(((2*b*c)^2 - (b^2 + c^2 - d^2)^2)/3) / d. - Jinyuan Wang, Feb 19 2022

Example

For the 1st triple in A336328, i.e., (57, 65, 73), we get A336329(1) = FA + FB + FC = 325/7 + 264/7 + 195/7 = 112, hence A351477(1) = 7 and a(1) = 325.

Prog

(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(d=6*(a^2*b^2+b^2*c^2+c^2*a^2)-3*(a^4+b^4+c^4)) && issquare(d=(a^2+b^2+c^2+sqrtint(d))/2), d = sqrtint(d); print1(numerator(sqrtint(((2*b*c)^2 - (b^2 + c^2 - d^2)^2)/3)/d), ", "); ); ); ); ); } \\ Michel Marcus, Mar 02 2022

Crossrefs

Cf. A336328 (primitive triples), A336329 (FA + FB + FC), A336330 (smallest side), A336331 (middle side), A336332 (largest side), A336333 (perimeter), this sequence (FA numerator), A351802 (FB numerator), A351803 (FC numerator), A351477 (common denominator of FA, FB, FC), A351476 (other 'FA + FB + FC').

Sequence in context: A025313 A025286 A025304 * A160580 A158272 A183645

Adjacent sequences: A351798 A351799 A351800 * A351802 A351803 A351804

Keyword

nonn

Author

Bernard Schott, Feb 19 2022

Extensions

More terms from Jinyuan Wang, Feb 19 2022

Status

approved