A096467 Numbers that can be the longest side of a primitive Heronian triangle.

5, 6, 8, 13, 15, 17, 20, 21, 24, 25, 26, 28, 29, 30, 35, 36, 37, 39, 40, 41, 42, 44, 45, 48, 50, 51, 52, 53, 55, 56, 58, 60, 61, 63, 65, 66, 68, 69, 70, 73, 74, 75, 77, 80, 82, 85, 87, 88, 89, 90, 91, 92, 93, 95, 96, 97, 100, 101, 102, 104, 105, 106, 109, 110, 111, 112, 113

Offset

1,1

Comments

Here a primitive Heronian triangle has integer sides a,b,c with gcd(a,b,c) = 1 and integral area. Note that all primes of the form 4k+1 are in this sequence. It appears that a prime of the form 4k+3 is never the longest side of a Heronian triangle. Cheney's article contains many theorems about these triangles.

Links

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 240 terms from Vincenzo Librandi)

Wm. Fitch Cheney, Jr., Heronian Triangles, Amer. Math. Monthly, Vol. 36, No. 1 (Jan 1929), 22-28.

Eric Weisstein's World of Mathematics, Heronian Triangle

Example

5 is on this list because the triangle with sides 3, 4, 5 has integral area.

Mathematica

nn=150; lst={}; Do[s=(a+b+c)/2; If[IntegerQ[s] && GCD[a, b, c]==1, area2=s(s-a)(s-b)(s-c); If[area2>0 && IntegerQ[Sqrt[area2]], AppendTo[lst, a]]], {a, nn}, {b, a}, {c, b}]; Union[lst]

Crossrefs

Cf. A083875 (area/6 of primitive Heronian triangles), A096468 (perimeter of primitive Heronian triangles).

Cf. A239246, A306626.

Sequence in context: A276374 A184803 A331264 * A334120 A323041 A105830

Adjacent sequences: A096464 A096465 A096466 * A096468 A096469 A096470

Keyword

nonn

Author

T. D. Noe, Jun 22 2004

Status

approved