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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
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
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
KEYWORD
nonn
AUTHOR
Michel Lagneau, Sep 16 2013
STATUS
approved