cp's OEIS Frontend

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.

%I A336332 #23 Feb 12 2025 09:26:51
%S A336332 73,95,152,205,208,280,296,343,361,387,407,437,469,473,485,624,728,
%T A336332 931,1016,1273,1311,1313,1368,1387,1443,1457,1463,1469,1477,1519,1560,
%U A336332 1591,1687,1895,2015,2045,2045,2085,2197,2231,2289,2347,2363,2416,2465,2553,2728,2821,2923
%N 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.
%C A336332 Inspired by Project Euler, Problem 143 (see link).
%C A336332 This sequence is increasing because triples are in increasing order of largest side.
%C A336332 For the corresponding primitive triples and miscellaneous properties and references, see A336328.
%C A336332 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.
%D A336332 Martin Gardner, Mathematical Circus, Elegant triangles, First Vintage Books Edition, 1979, p. 65
%H A336332 Project Euler, <a href="https://projecteuler.net/problem=143">Problem 143 - Investigating the Torricelli point of a triangle</a>.
%F A336332 a(n) = A336328(n, 3).
%e A336332 a(36) = a(37) = 2045 is the smallest largest side that appears twice because:
%e A336332    (1023, 1387, 2045) is a triple with FA+FB+FC = 2408, and
%e A336332    (1051, 1744, 2045) is a triple with FA+FB+FC = 2709.
%Y A336332 Cf. A336328 (triples), A336329 (FA + FB + FC), A336330 (smallest side), A336331 (middle side), this sequence (largest side), A336333 (perimeter).
%Y A336332 Cf. A072052 (largest sides: primitives and multiples), A333391.
%K A336332 nonn
%O A336332 1,1
%A A336332 _Bernard Schott_, Jul 20 2020