OFFSET
1,1
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 *
....................
LINKS
Eric W. Weisstein, Heron's Formula
EXAMPLE
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.
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Lagneau, Sep 28 2013
EXTENSIONS
Extended by T. D. Noe, Sep 30 2013
Missing term 2970 from Giovanni Resta, Mar 08 2017
STATUS
approved