A227112 Numbers n such that there exist two primes p and q where the area A of the triangle of sides (n, p, q) is an integer.

4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 26, 30, 38, 40, 42, 50, 56, 60, 68, 70, 78, 80, 90, 96, 100, 102, 104, 110, 120, 130, 144, 148, 150, 156, 160, 170, 174, 180, 182, 198, 210, 224, 234, 240, 286, 290, 300, 312, 350, 360, 370, 390, 400, 440, 510, 520, 548

Offset: 1

Author

Michel Lagneau, Oct 02 2013

Keywords

• nonn

Comments

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 |

...

Examples

			12 is in the sequence because the triangle (12, 5, 13) => semiperimeter s = (12+5+13)/2 = 15, and A = sqrt(15*(15-12)*(15-5)*(15-13))= 30, with 5 and 13 prime numbers.
		

Crossrefs

Cf. A229159, A229746.

Programs

Mathematica

area[a_, b_, c_] := Module[{s = (a + b + c)/2, a2}, a2 = s (s - a) (s - b) (s - c); If[a2 < 0, 0, Sqrt[a2]]]; goodQ[a_, b_, c_] := Module[{ar = area[a, b, c]}, ar > 0 && IntegerQ[ar]]; nn = 300; t = {}; ps = Prime[Range[2, nn]]; mx = 3*ps[[-1]]; Do[If[p <= q && goodQ[p, q, e], aa = area[p, q, e]; If[aa <= mx, AppendTo[t, e]]], {p, ps}, {q, ps}, {e, q - p + 2, p + q - 2, 2}]; t = Union[t] (* program from T. D. Noe adapted for this sequence - see A229746 *)