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Search: id:A096468
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| A096468 |
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Numbers n that can be the perimeter of a primitive Heronian triangle. |
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+0
2
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| 12, 16, 18, 30, 32, 36, 40, 42, 44, 48, 50, 54, 56, 60, 64, 66, 68, 70, 72, 76, 78, 80, 84, 90, 96, 98, 100, 104, 108, 110, 112, 114, 120, 126, 128, 130, 132, 136, 140, 144, 150, 152, 154, 156, 160, 162, 164, 168, 170, 172, 174, 176, 180, 182, 186, 190, 192, 196
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Here a primitive Heronian triangle has integer sides a,b,c with GCD(a,b,c) = 1 and integral area. The perimeter is always even. Cheney's article contains many theorems about these triangles.
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REFERENCES
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Wm. Fitch Cheney, Jr., Heronian Triangles, Amer. Math. Monthly, Vol. 36, No. 1 (Jan 1929), 22-28.
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LINKS
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Eric Weisstein's World of Mathematics, Heronian Triangle
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EXAMPLE
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12 is on this list because the triangle with sides 3, 4, 5 has integral area and perimeter 12.
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MATHEMATICA
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nn=150; lst={}; Do[s=(a+b+c)/2; If[IntegerQ[s] && GCD[a, b, c]==1, area2=s(s-a)(s-b)(s-c); If[area2>0 && IntegerQ[Sqrt[area2]], AppendTo[lst, 2s]]], {a, nn}, {b, a}, {c, b}]; Union[lst]
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CROSSREFS
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Cf. A070138 (number of primitive Heronian triangles having perimeter n), A083875 (area/6 of primitive Heronian triangles), A096467 (longest side of primitive Heronian triangles).
Adjacent sequences: A096465 A096466 A096467 this_sequence A096469 A096470 A096471
Sequence in context: A089021 A112548 A032620 this_sequence A054281 A070329 A064695
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KEYWORD
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nonn
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AUTHOR
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T. D. Noe (noe(AT)sspectra.com), Jun 22 2004
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