login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A226453 Integer areas of integer-sided triangles where at least one side is of prime length. 3
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 (list; graph; refs; listen; history; text; internal format)
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 whose 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 7 19:23 EDT 2021. Contains 343652 sequences. (Running on oeis4.)