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

1,1

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 *

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

Links

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

Example

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.

Mathematica

nn=1000; lst={}; Do[s=(a+b+c)/2; If[IntegerQ[s], area2=s (s-a) (s-b) (s-c); If[0<area2<=nn^2 && (PrimeQ[a] || PrimeQ[b] || PrimeQ[c]) && IntegerQ[Sqrt[area2]], AppendTo[lst, Sqrt[area2]]]], {a, nn}, {b, a}, {c, b}]; Union[lst]

Crossrefs

Cf. A188158.

Sequence in context: A247145 A188158 A061822 * A307225 A261476 A119840

Adjacent sequences: A226450 A226451 A226452 * A226454 A226455 A226456

Keyword

nonn

Author

Michel Lagneau, Sep 16 2013

Status

approved