A070142 - OEIS

%I #13 Jun 12 2017 09:41:29

%S 17,39,52,116,212,252,269,368,370,372,375,493,561,587,659,839,850,862,

%T 957,972,1156,1186,1196,1204,1297,1582,1599,1629,1912,1920,1955,1971,

%U 1988,2115,2352,2555,2574,2713,2774,2778,2790

%N Numbers n such that [A070080(n), A070081(n), A070082(n)] is an integer triangle with integer area.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HeronianTriangle.html">Heronian Triangle</a>.

%H R. Zumkeller, <a href="/A070080/a070080.txt">Integer-sided triangles</a>

%e a(2)=39: [A070080(39), A070081(39), A070082(39)] = [5,5,6], area^2 = s*(s-5)*(s-5)*(s-6) with s=A070083(39)/2=(5+5+6)/2=8, area^2=8*3*3*2=16*9 is an integer square, therefore A070086(39)=area=4*3=12.

%t maxPerim = 100; maxSide = Floor[(maxPerim - 1)/2]; order[{a_, b_, c_}] := (a + b + c)*maxPerim^3 + a*maxPerim^2 + b*maxPerim + c; triangles = Reap[ Do[ If[ a + b + c <= maxPerim && c - b < a < c + b && b - a < c < b + a && c - a < b < c + a, Sow[{a, b, c}]], {a, 1, maxSide}, {b, a, maxSide}, {c, b, maxSide}]][[2, 1]]; stri = Sort[ triangles, order[#1] < order[#2]&]; area[{a_, b_, c_}] := With[{p = (a + b + c)/2}, Sqrt[p*(p - a)*(p - b)*(p - c)]]; Position[ stri, tri_ /; IntegerQ[area[tri]]] // Flatten (* _Jean-François Alcover_, Feb 22 2013 *)

%Y Cf. A070088, A070149, A070143, A070144, A070145, A051516, A069594, A069597, A070136, A070137, A070209.

%K nonn

%O 1,1

%A _Reinhard Zumkeller_, May 05 2002