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A229746
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Integer areas of integer-sided triangles where two sides are of prime length.
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3
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6, 12, 30, 60, 66, 72, 114, 120, 180, 210, 240, 330, 336, 360, 396, 420, 456, 660, 756, 780, 840, 900, 984, 1116, 1200, 1248, 1260, 1290, 1320, 1584, 1590, 1680, 1710, 1716, 1770, 1800, 1980, 2100, 2310, 2400, 2460, 2496, 2520, 2604, 2640, 2940, 2970, 3060, 3120
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OFFSET
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1,1
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COMMENTS
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Subset of A188158. The length of the third side is an even composite number because the perimeter is always even.
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.
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 *
* 30 * 5 * 12 * 13 *
* 60 * 10 * 13 * 13 *
* 66 * 11 * 13 * 20 *
* 72 * 5 * 29 * 30 *
* 114 * 19 * 20 * 37 *
* 120 * 16 * 17 * 17 *
* 120 * 17 * 17 * 30 *
* 180 * 13 * 30 * 37 *
....................
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LINKS
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EXAMPLE
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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, with 19 and 37 prime numbers.
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MATHEMATICA
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area[a_, b_, c_] := Module[{s = (a + b + c)/2, a2}, a2 = s (s - a) (s - b) (s - c); If[a2 < 0, 0, Sqrt[a2]]]; goodQ[a_, b_, c_] := Module[{ar = area[a, b, c]}, ar > 0 && IntegerQ[ar]]; nn = 80; t = {}; ps = Prime[Range[2, nn]]; mx = 3*ps[[-1]]; Do[If[p <= q && goodQ[p, q, e], aa = area[p, q, e]; If[aa <= mx, AppendTo[t, aa]]], {p, ps}, {q, ps}, {e, q - p + 2, p + q - 2, 2}]; t = Union[t] (* T. D. Noe, Oct 01 2013 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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