cp's OEIS Frontend

Subset of A188158.

The areas of the triangles (a,b,c) are given by Heron's formula, A = sqrt(s(s-a)(s-b)(s-c)), where its side lengths are a, b, c and semiperimeter s = (a+b+c)/2.

The areas A of the primitive triangles of sides (a,b,c) are 120, 168, 300, 360, 4680, ...

The areas of the nonprimitive triangle of sides (a*p^2, b*p^2, c*p^2) are in the sequence with the value A*p^4.

It is possible to find integer-sided triangles having two square sides, for example:

a(2) = 168 with sides (25,25,48) and (14,25,25);

a(3) = 300 with sides (25,25,30) and (25,25,40);

a(14) = 14280 with sides (169,169,238), (169,169,240), (100,289,291).

The following table gives the first values (A, a, b, c):

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

| A | a | b | c |

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

| 120 | 16 | 25 | 39 |

| 168 | 14 | 25 | 25 |

| 300 | 25 | 25 | 40 |

| 360 | 25 | 29 | 36 |

| 1920 | 64 | 100 | 156 |

| 2016 | 64 | 225 | 287 |

| 2688 | 100 | 100 | 192 |

| 4680 | 74 | 169 | 225 |

| 4800 | 100 | 100 | 160 |

| 5760 | 100 | 116 | 144 |

| 9720 | 144 | 225 | 351 |

| 10140 | 169 | 169 | 312 |

| 13608 | 225 | 225 | 432 |

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

n is an even composite number because the perimeter of the triangle (n, p, q) is always even. The corresponding areas are {6, 12, 12, 60, 30, 120, 360, 66, ...}

The area is given by Heron's formula A = sqrt(s(s-n)(s-p)(s-q)) where the semiperimeter s = (n + p + q)/2.

The following table gives the first values (n, A, p, q).

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

| n | A | p | q |

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

| 4 | 6 | 3 | 5 |

| 6 | 12 | 5 | 5 |

| 8 | 12 | 5 | 5 |

| 10 | 60 | 13 | 13 |

| 12 | 30 | 5 | 13 |

| 16 | 120 | 17 | 17 |

| 18 | 360 | 41 | 41 |

| 20 | 66 | 11 | 13 |

...

A001223(n) = A000040(n+1) - A000040(n) = prime(n+1) - prime(n).

The Mathematica program examines all triangles with n <= 10^8.

The sequence a(n) is a subsequence of A188158, and the lengths of the sides are even.

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.

a(n) == 0 mod 24 => {b(n)} = {a(n)/24} = {1, 4, 5, 6, 7, 9, 10, 11, 14, 15, 16, 18, 19, 20, 21, 22, 24, 25, 26, 28, 30, 33, 34, 35, 36, 39, 40, 42, 44, 45, 49, 51, 54, 55, 56, 60, 63, 64, 65, 70, 72, ...}. It seems that the primes > 19 are not in {b(n)}.

For the same area, the number of distinct triangles is not always unique; for example, the area 336 can be obtained with triangle (30, 28, 26) starting from prime 461983 and also from triangle (34, 20, 42) starting from prime 2473663 (Giovanni Resta, Mar 08 2017).

The following table gives the first values (A, m, sides of the triangles) where A is the area of the triangles and m is the smallest value generating A.

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

| A | m | A001223(m)| A001223(m+1)| A001223(m+2)|

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

| 24 | 123 | 6 | 8 | 10 |

| 96 | 3935 | 16 | 20 | 12 |

| 120 | 8101 | 10 | 26 | 24 |

| 144 | 13097 | 34 | 18 | 20 |

| 168 | 12226 | 30 | 40 | 14 |

| 216 | 9864 | 24 | 18 | 30 |

| 240 | 102715 | 58 | 50 | 12 |

| 264 | 98259 | 22 | 26 | 40 |

| 336 | 38604 | 30 | 28 | 26 |

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