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A352360 Three-column array giving list of primitive triples for integer-sided triangles with A < B < C < 2*Pi/3 and such that FA, FB, FC are also integers where F is the Fermat point of the triangle.

%I A352360 #41 Apr 03 2023 14:35:51
%S A352360 399,455,511,511,616,665,1591,5439,5624,35941,47544,58015,8827,16835,
%T A352360 18928,36741,73151,92680,16219,94335,97976,1235,4056,4459,12728,13545,
%U A352360 15523,14744,33271,37539,13889,16856,17501,1911,4901,5681,196935,320624,324079,9435,12691,17501,22477,37128,44135
%N A352360 Three-column array giving list of primitive triples for integer-sided triangles with A < B < C < 2*Pi/3 and such that FA, FB, FC are also integers where F is the Fermat point of the triangle.
%C A352360 Inspired by Project Euler, Problem 143 (see link) where such a triangle is called a Torricelli triangle.
%C A352360 Differs from A336328 where FA + FB + FC is an integer, but FA, FB and FC are fractions. _Jinyuan Wang_ has found that the 37th and 58th triples are the first triples for which the common denominator of these fractions is 1 (A351477).
%C A352360 Each triple (a, b, c) is in increasing order, and the triples are displayed in the same increasing order of the corresponding triples in A336328 (see formulas).
%C A352360 +-------+-------+-------+---------+--------+-------+-------+--------+--------+
%C A352360 |    a  |   b   |   c   |gcd(a,b,c)|   FA  |   FB  |   FC  |    d   | a+b+c  |
%C A352360 +-------+-------+-------+----------+-------+-------+-------+--------+--------+
%C A352360 |   399 |   455 |   511 |     7    |   325 |   264 |   195 |    784 |   1365 |
%C A352360 |   511 |   616 |   665 |     7    |   440 |   325 |   264 |   1029 |   1792 |
%C A352360 |  1591 |  5439 |  5624 |    37    |  5016 |  1064 |   765 |   6845 |  12654 |
%C A352360 | 35941 | 47544 | 58015 |   283    | 39360 | 27265 | 13464 |  80089 | 141500 |
%C A352360 |  8827 | 16835 | 18928 |    91    | 14800 |  6528 |  3515 |  24843 |  44590 |
%C A352360 | 36741 | 73151 | 92680 |   331    | 70720 | 34200 |  4641 | 109561 | 202572 |
%C A352360 | 16219 | 94335 | 97976 |   331    | 91200 | 12376 |  5985 | 109561 | 208530 |
%C A352360 |  1235 |  4056 |  4459 |    13    |  3864 |  1015 |   360 |   5239 |   9750 |
%C A352360 | 12728 | 13545 | 15523 |    43    |  9405 |  8512 |  6120 |  24037 |  41796 |
%C A352360 | 14744 | 33271 | 37539 |    97    | 30429 | 11520 |  5096 |  47045 |  87554 |
%C A352360 ..............................................................................
%C A352360 The sequences with terms of this table are listed in Crossrefs section; here, d = FA + FB + FC. The perimeter corresponding to n-th triple a+b+c = A336333(n) * A351477(n).
%H A352360 Project Euler, <a href="https://projecteuler.net/problem=143">Problem 143 - Investigating the Torricelli point of a triangle</a>.
%F A352360 a(3n-2) = A336328(3n-2) * A351477(n), a(3n-1) = A336328(3n-1) * A351477(n), a(3n) = A336328(3n) * A351477(n), for n >= 1.
%e A352360 The array begins:
%e A352360     399,   455,   511;
%e A352360     511,   616,   665;
%e A352360    1591,  5439,  5624;
%e A352360   35941, 47544, 58015;
%e A352360    8827, 16835, 18928;
%e A352360   36741, 73151, 92680;
%e A352360   .....................
%e A352360 For 1st triple (399, 455, 511) with gcd(399, 455, 511) = 7, we get FA = 325, FB = 264 and FC = 195. This smallest triangle such that a, b, c, FA, FB, FC are all integers is the example proposed in Project Euler's link.
%Y A352360 Cf. A336328.
%Y A352360 Cf. A351477 (gcd(a,b,c)), A351801 (FA), A351802 (FB), A351803 (FC), A351476 (FA+FB+FC).
%K A352360 nonn,tabf
%O A352360 1,1
%A A352360 _Bernard Schott_, Mar 17 2022