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.

399, 455, 511, 511, 616, 665, 1591, 5439, 5624, 35941, 47544, 58015, 8827, 16835, 18928, 36741, 73151, 92680, 16219, 94335, 97976, 1235, 4056, 4459, 12728, 13545, 15523, 14744, 33271, 37539, 13889, 16856, 17501, 1911, 4901, 5681, 196935, 320624, 324079, 9435, 12691, 17501, 22477, 37128, 44135

Offset

1,1

Comments

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

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).

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).

+-------+-------+-------+---------+--------+-------+-------+--------+--------+

| a | b | c |gcd(a,b,c)| FA | FB | FC | d | a+b+c |

+-------+-------+-------+----------+-------+-------+-------+--------+--------+

| 399 | 455 | 511 | 7 | 325 | 264 | 195 | 784 | 1365 |

| 511 | 616 | 665 | 7 | 440 | 325 | 264 | 1029 | 1792 |

| 1591 | 5439 | 5624 | 37 | 5016 | 1064 | 765 | 6845 | 12654 |

| 35941 | 47544 | 58015 | 283 | 39360 | 27265 | 13464 | 80089 | 141500 |

| 8827 | 16835 | 18928 | 91 | 14800 | 6528 | 3515 | 24843 | 44590 |

| 36741 | 73151 | 92680 | 331 | 70720 | 34200 | 4641 | 109561 | 202572 |

| 16219 | 94335 | 97976 | 331 | 91200 | 12376 | 5985 | 109561 | 208530 |

| 1235 | 4056 | 4459 | 13 | 3864 | 1015 | 360 | 5239 | 9750 |

| 12728 | 13545 | 15523 | 43 | 9405 | 8512 | 6120 | 24037 | 41796 |

| 14744 | 33271 | 37539 | 97 | 30429 | 11520 | 5096 | 47045 | 87554 |

..............................................................................

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).

Links

Table of n, a(n) for n=1..45.

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

Formula

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.

Example

The array begins:
    399,   455,   511;
    511,   616,   665;
   1591,  5439,  5624;
  35941, 47544, 58015;
   8827, 16835, 18928;
  36741, 73151, 92680;
  .....................
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.

Crossrefs

Cf. A336328.

Cf. A351477 (gcd(a,b,c)), A351801 (FA), A351802 (FB), A351803 (FC), A351476 (FA+FB+FC).

Sequence in context: A274446 A253598 A046013 * A176911 A202158 A126231

Adjacent sequences: A352357 A352358 A352359 * A352361 A352362 A352363

Keyword

nonn,tabf

Author

Bernard Schott, Mar 17 2022

Status

approved