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A227112
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Numbers n such that there exist two primes p and q where the area A of the triangle of sides (n, p, q) is an integer.
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0
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4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 26, 30, 38, 40, 42, 50, 56, 60, 68, 70, 78, 80, 90, 96, 100, 102, 104, 110, 120, 130, 144, 148, 150, 156, 160, 170, 174, 180, 182, 198, 210, 224, 234, 240, 286, 290, 300, 312, 350, 360, 370, 390, 400, 440, 510, 520, 548
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OFFSET
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1,1
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COMMENTS
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n is an even composite number because the perimeter of the triangle (n, p, q) is always even. The corresponding areas are {6, 12, 12, 60, 30, 120, 360, 66, ...}
The area is given by Heron's formula A = sqrt(s(s-n)(s-p)(s-q)) where the semiperimeter s = (n + p + q)/2.
The following table gives the first values (n, A, p, q).
+----+-----+----+----+
| n | A | p | q |
+----+-----+----+----+
| 4 | 6 | 3 | 5 |
| 6 | 12 | 5 | 5 |
| 8 | 12 | 5 | 5 |
| 10 | 60 | 13 | 13 |
| 12 | 30 | 5 | 13 |
| 16 | 120 | 17 | 17 |
| 18 | 360 | 41 | 41 |
| 20 | 66 | 11 | 13 |
...
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LINKS
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EXAMPLE
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12 is in the sequence because the triangle (12, 5, 13) => semiperimeter s = (12+5+13)/2 = 15, and A = sqrt(15*(15-12)*(15-5)*(15-13))= 30, with 5 and 13 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 = 300; 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, e]]], {p, ps}, {q, ps}, {e, q - p + 2, p + q - 2, 2}]; t = Union[t] (* program from T. D. Noe adapted for this sequence - see A229746 *)
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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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