cp's OEIS Frontend

A096468 Perimeters of primitive Heronian triangles.

%I A096468 #22 Feb 16 2025 08:32:53
%S A096468 12,16,18,30,32,36,40,42,44,48,50,54,56,60,64,66,68,70,72,76,78,80,84,
%T A096468 90,96,98,100,104,108,110,112,114,120,126,128,130,132,136,140,144,150,
%U A096468 152,154,156,160,162,164,168,170,172,174,176,180,182,186,190,192,196
%N A096468 Perimeters of primitive Heronian triangles.
%C A096468 Here a primitive Heronian triangle has integer sides a,b,c with GCD(a,b,c) = 1 and integral area. The perimeter is always even. Cheney's article contains many theorems about these triangles.
%H A096468 Peter Kagey and Giovanni Resta, <a href="/A096468/b096468.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Peter Kagey)
%H A096468 Wm. Fitch Cheney, Jr., <a href="http://www.jstor.org/stable/2300173">Heronian Triangles</a>, Amer. Math. Monthly, Vol. 36, No. 1 (Jan 1929), 22-28.
%H A096468 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HeronianTriangle.html">Heronian Triangle</a>
%e A096468 12 is on this list because the triangle with sides 3, 4, 5 has integral area and perimeter 12.
%t A096468 nn=150; lst={}; Do[s=(a+b+c)/2; If[IntegerQ[s] && GCD[a, b, c]==1, area2=s(s-a)(s-b)(s-c); If[area2>0 && IntegerQ[Sqrt[area2]], AppendTo[lst, 2s]]], {a, nn}, {b, a}, {c, b}]; Union[lst]
%Y A096468 Cf. A070138 (number of primitive Heronian triangles having perimeter n), A083875 (area/6 of primitive Heronian triangles), A096467 (longest side of primitive Heronian triangles).
%K A096468 nonn
%O A096468 1,1
%A A096468 _T. D. Noe_, Jun 22 2004
%E A096468 Name changed by _Wesley Ivan Hurt_, May 17 2020