A226453 Integer areas of integer-sided triangles where at least one side is of prime length.

6, 12, 24, 30, 36, 42, 60, 66, 72, 84, 90, 114, 120, 126, 132, 156, 180, 204, 210, 216, 234, 240, 252, 264, 270, 288, 300, 306, 330, 336, 360, 390, 396, 420, 456, 462, 504, 510, 522, 528, 546, 570, 624, 630, 660, 684, 690, 714, 720, 756, 780, 798, 840, 864

Offset: 1

Author

Michel Lagneau, Sep 16 2013

Keywords

• nonn

Comments

Subset of A188158.

The area of the triangles (a,b,c) are given by Heron's formula A = sqrt(s(s-a)(s-b)(s-c)) where its side lengths are a, b, c and semiperimeter s = (a+b+c)/2.

There exist triangles where two distinct integer sides are a prime number, for example:

a(n) = 6 with sides (3,4,5);

a(n) = 30 with sides (5,12,13);

a(n) = 66 with sides (11,13,20);

a(n) = 72 with sides (5,29,30);

a(n) = 114 with sides (19,20,37).

The following table gives the first values (A, a, b, c):

**********************

* A * a * b * c *

**********************

* 6 * 3 * 4 * 5 *

* 12 * 5 * 5 * 6 *

* 12 * 5 * 5 * 8 *

* 24 * 4 * 13 * 15 *

* 30 * 5 * 12 * 13 *

* 36 * 3 * 25 * 26 *

* 36 * 9 * 10 * 17 *

* 42 * 7 * 15 * 20 *

* 60 * 6 * 25 * 29 *

* 60 * 8 * 15 * 17 *

* 60 * 10 * 13 * 13 *

* 60 * 13 * 13 * 24 *

* 66 * 11 * 13 * 20 *

* 72 * 5 * 29 * 30 *

......................

Examples

			114 is in the sequence because the triangle (19, 20, 37) => semiperimeter s = (19+20+37)/2 = 38, and A = sqrt(38*(38-19)*(38-20)*(38-37)) = 114.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A188158.

Programs

Mathematica

nn=1000; lst={}; Do[s=(a+b+c)/2; If[IntegerQ[s], area2=s (s-a) (s-b) (s-c); If[0