cp's OEIS Frontend

Subset of A226453.

The corresponding primes are: 3, 3, 5, 5, 7, 7, 11, 13, 11, 13, 17, 19, 17, 23, 19, 23, 29, 31, 29, 37, 31, 41, 43, 37, 47, 41, 53, 43, 59, 47, 61, 53, 67, 71, 73, 59, 61, 79, 83, 67, 89, ...

The area A of a triangle whose sides have lengths a, b, and c is given by Heron's formula: A = sqrt(s*(s-a)*(s-b)*(s-c)), where s = (a+b+c)/2.

The altitudes of a triangle with sides length a, b, c and area A have length given by Ha = 2*A/a, Hb = 2*A/b, Hc = 2*A/c.

Properties of this sequence:

- The sequence is infinite (see the formula below);

- The prime altitude of a triangle is the greatest prime divisor of a(n) (the proof is easy if we observe the formula);

- There exists two subsets of numbers included into a(n):

Case (i): A subset with right triangles (a,b,c) where a^2 + b^2 = c^2 with area a1(n) = {6, 30, 84, 330, 546, 1224, ...}. The lengths of the prime altitudes are Ha or Hb = a = p. The sides are of the form (p, q, q+1) with p = sqrt(2*q+1) => the sides are equal to (p, (p^2-1)/2, (p^2+1)/2) and a(n) = (p^3-p)/4.

Case (ii): A subset with isosceles triangles formed by two right triangles of the sequence. So, the areas are a2(n) = {12, 60, 168, 660, 1092, 2448, ...} = 2*a1(n). The sides are of the form (a, a, 2*(a-1)) = ((p^2+1)/2, (p^2+1)/2, p^2-1) and Ha = sqrt(2*a-1) = p, a2(n) = 2*a1(n) = (p^3-p)/2.

We did not find a class of non-isosceles and non-right triangles (a, b, c) whose three altitudes include one of prime length.

The following table gives the first values (A, a, b, c, Ha, Hb, Hc) where A is the integer area, a, b, c are the sides and Ha <= Hb <= Hc are the altitudes.

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

| A | a | b | c | Ha | Hb | Hc |

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

| 6 | 3 | 4 | 5 | 12/5 | 3 | 4 |

| 12 | 5 | 5 | 8 | 3 | 24/5 | 24/5 |

| 30 | 5 | 12 | 13 | 5 | 60/13 | 12 |

| 60 | 13 | 13 | 24 | 5 | 120/13 | 120/13 |

| 84 | 7 | 24 | 25 | 168/25 | 7 | 24 |

| 168 | 25 | 25 | 48 | 7 | 336/25 | 336/25 |

| 330 | 11 | 60 | 61 | 660/61 | 11 | 60 |

| 546 | 13 | 84 | 85 | 1092/85 | 13 | 84 |

| 660 | 61 | 61 | 120 | 11 | 1320/61 | 1320/61 |

| 1092 | 85 | 85 | 168 | 13 | 2184/85 | 2184/85 |

| 1224 | 17 | 144 | 145 | 2448/145 | 17 | 144 |

| 1710 | 19 | 180 | 181 | 3420/181 | 19 | 180 |

| 2448 | 145 | 145 | 288 | 4896/145 | 4896/145 | 17 |

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