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A230195
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Integer areas A of the triangles such that A and the sides are integers, and the length of the inradius is a prime number.
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1
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24, 30, 36, 42, 48, 54, 60, 66, 84, 96, 114, 120, 126, 150, 156, 198, 210, 270, 294, 330, 336, 390, 420, 462, 504, 510, 546, 570, 630, 714, 726, 756, 810, 840, 924, 930, 1008, 1014, 1056, 1134, 1386, 1428, 1554, 1638, 1680, 1716, 1734, 1848, 1890, 1950, 2016
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OFFSET
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1,1
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COMMENTS
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The corresponding inradii r are 2, 2, 2, 2, 3, 3, 2, 3, 3, 3, 3, 3, 3, 5, 3, 3, 3, 5, 7, 5, 7, 5, 7, 7, 7, 5, 7, 5, ...
The area A of a triangle whose sides have lengths a, b, and c is given by Heron's formula: A = sqrt(s*(s-a)*(s-b)*(s-c)), where s = (a+b+c)/2. The inradius r is given by r = A/s.
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LINKS
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EXAMPLE
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24 is in the sequence because for (a, b, c) = (6, 8, 10) => s =(6 + 8 + 10)/2 = 12; A = sqrt(12*(12-6)*(12-8)*(12-10)) = sqrt(576)= 24; r = A/s = 2 is prime.
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MATHEMATICA
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nn = 1500; lst = {}; Do[s = (a + b + c)/2; If[IntegerQ[s], area2 = s (s - a) (s - b) (s - c); If[0 < area2 <= nn^2 && IntegerQ[Sqrt[area2]] && PrimeQ[Sqrt[area2]/s], AppendTo[lst, Sqrt[area2]]]], {a, nn}, {b, a}, {c, b}]; Union[lst]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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