A229746 Integer areas of integer-sided triangles where two sides are of prime length.

6, 12, 30, 60, 66, 72, 114, 120, 180, 210, 240, 330, 336, 360, 396, 420, 456, 660, 756, 780, 840, 900, 984, 1116, 1200, 1248, 1260, 1290, 1320, 1584, 1590, 1680, 1710, 1716, 1770, 1800, 1980, 2100, 2310, 2400, 2460, 2496, 2520, 2604, 2640, 2940, 2970, 3060, 3120

Offset: 1

Author

Michel Lagneau, Sep 28 2013

Keywords

• nonn

Comments

Subset of A188158. The length of the third side is an even composite number because the perimeter is always even.

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

***********************

* A * a * b * c *

***********************

* 6 * 3 * 4 * 5 *

* 12 * 5 * 5 * 6 *

* 12 * 5 * 5 * 8 *

* 30 * 5 * 12 * 13 *

* 60 * 10 * 13 * 13 *

* 66 * 11 * 13 * 20 *

* 72 * 5 * 29 * 30 *

* 114 * 19 * 20 * 37 *

* 120 * 16 * 17 * 17 *

* 120 * 17 * 17 * 30 *

* 180 * 13 * 30 * 37 *

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

Examples

			114 is in the sequence because the triangle (19, 20, 37) => semiperimeter s = (19+20+37)/2 = 38, and A = sqrt(38*(38-19)*(38-20)*(38-37)) = 114, with 19 and 37 prime numbers.
		

Links

• Eric W. Weisstein, Heron's Formula

Crossrefs

Cf. A188158, A229159.

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 = 80; 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, aa]]], {p, ps}, {q, ps}, {e, q - p + 2, p + q - 2, 2}]; t = Union[t] (* T. D. Noe, Oct 01 2013 *)

Extensions

Extended by T. D. Noe, Sep 30 2013

Missing term 2970 from Giovanni Resta, Mar 08 2017