A068969 Areas of integer Heronian triangles [A068967(n), prime(A068967(n)), A068968(n)].
6, 210, 528, 1680, 1800, 5304, 6930, 3150, 4650, 21000, 32760, 69342, 53550, 2170560, 2200200, 2501070, 646800, 4777080, 4796550, 11865840, 12243840, 12863760, 18064200, 30510480, 3232320, 66023100, 70691400, 47977380, 144357720, 185560830, 156128700, 320843040
Offset: 1
Keywords
Examples
a(5) = 37: [37, A000040(37), A068968(5)] = [37,157,130], with s = (36+157+130)/2 = 162: Area^2 = s*(s-37)*(s-157)*(s-130) = 162*125*5*32 = 3240000 = 1800*1800, therefore a(5) = 1800.
Links
- Giovanni Resta, Table of n, a(n) for n = 1..120
- Eric Weisstein's World of Mathematics, Heronian Triangle.
- Eric Weisstein's World of Mathematics, Heron's Formula.
Programs
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Mathematica
area[{a_, b_, c_}] := Block[{s = (a + b + c)/2}, Sqrt[s (s-a) (s-b) (s-c)]]; zz[n_] := Block[{s, p = Prime[n], t, z, A}, s = Solve[(n^2 - 2 n p + p^2 - z) (z - n^2 - 2 n p - p^2) == t^2 && z>0 && t>0, {t, z}, Integers]; If[s == {}, {}, z = Sort@ Select[Sqrt[z /. s], IntegerQ]]; Select[Table[{n, p, e}, {e, z}], IntegerQ[A = area[#]] && A > 0 &]]; area /@ Join @@ Parallelize@ Array[zz, 4300] (* Giovanni Resta, Apr 20 2020 *)
Extensions
Erroneous term 25070627 removed and more terms from Giovanni Resta, Apr 20 2020