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

1,1

Links

Giovanni Resta, Table of n, a(n) for n = 1..120

Eric Weisstein's World of Mathematics, Heronian Triangle.

Example

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.

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 *)

Crossrefs

Sequence in context: A297546 A257600 A211612 * A228887 A185592 A296280

Adjacent sequences: A068965 A068966 A068967 * A068969 A068970 A068971

Keyword

nonn

Author

Reinhard Zumkeller, Mar 30 2002

Extensions

Erroneous term 21254 removed and more terms from Giovanni Resta, Apr 20 2020

Status

approved