cp's OEIS Frontend

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

%I A229746 #23 Feb 16 2025 08:33:20
%S A229746 6,12,30,60,66,72,114,120,180,210,240,330,336,360,396,420,456,660,756,
%T A229746 780,840,900,984,1116,1200,1248,1260,1290,1320,1584,1590,1680,1710,
%U A229746 1716,1770,1800,1980,2100,2310,2400,2460,2496,2520,2604,2640,2940,2970,3060,3120
%N A229746 Integer areas of integer-sided triangles where two sides are of prime length.
%C A229746 Subset of A188158. The length of the third side is an even composite number because the perimeter is always even.
%C A229746 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.
%C A229746 The following table gives the first values (A, a, b, c):
%C A229746 ***********************
%C A229746 *   A *  a *  b *  c  *
%C A229746 ***********************
%C A229746 *   6 *  3 *  4 *  5  *
%C A229746 *  12 *  5 *  5 *  6  *
%C A229746 *  12 *  5 *  5 *  8  *
%C A229746 *  30 *  5 * 12 * 13  *
%C A229746 *  60 * 10 * 13 * 13  *
%C A229746 *  66 * 11 * 13 * 20  *
%C A229746 *  72 *  5 * 29 * 30  *
%C A229746 * 114 * 19 * 20 * 37  *
%C A229746 * 120 * 16 * 17 * 17  *
%C A229746 * 120 * 17 * 17 * 30  *
%C A229746 * 180 * 13 * 30 * 37  *
%C A229746 ....................
%H A229746 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/HeronsFormula.html">Heron's Formula</a>
%e A229746 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.
%t A229746 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 *)
%Y A229746 Cf. A188158, A229159.
%K A229746 nonn
%O A229746 1,1
%A A229746 _Michel Lagneau_, Sep 28 2013
%E A229746 Extended by _T. D. Noe_, Sep 30 2013
%E A229746 Missing term 2970 from _Giovanni Resta_, Mar 08 2017