A068967 Numbers that define integer Heronian triangles [a(n), prime(a(n)), A068968(n)] with area A068969(n).
3, 12, 25, 30, 37, 51, 60, 65, 65, 100, 116, 260, 273, 761, 761, 825, 884, 1092, 1476, 1695, 1695, 1752, 2415, 3048, 3552, 4045, 4134, 4264, 5429, 6357, 7267, 9316, 14027, 16095, 18850, 21576, 29925, 30261, 38164, 40562, 45982, 45982, 51805, 51805, 82000, 82000
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 A068969(5) = 1800.
Links
- Giovanni Resta, Table of n, a(n) for n = 1..120
- Eric Weisstein's World of Mathematics, Heronian Triangle.
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 &]]; (Join @@ Parallelize@ Array[zz, 4300])[[All, 1]] (* Giovanni Resta, Apr 20 2020 *)
Extensions
Erroneous term 2374 removed and more terms from Giovanni Resta, Apr 20 2020