A068968 Sides of integer Heronian triangles [A068967(n), prime(A068967(n)), a(n)] with area A068969(n).
4, 35, 120, 113, 130, 260, 319, 252, 372, 609, 595, 1905, 2020, 5712, 5790, 6148, 7735, 8789, 13651, 14126, 14576, 14725, 19856, 25907, 29603, 36366, 41425, 44325, 53628, 66164, 67830, 103525, 141044, 164396, 229121, 223805, 367336, 366860, 446405, 447855, 538135
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, 3]] (* Giovanni Resta, Apr 20 2020 *)
Extensions
Erroneous term 21254 removed and more terms from Giovanni Resta, Apr 20 2020