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A329536
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Integer areas of integer-sided triangles where the lengths of two of the sides are cubes.
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0
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480, 4200, 5148, 7500, 30720, 65520, 268800, 329472, 349920, 480000, 960960, 1684980, 1713660, 1884960, 1966080, 2413320, 2419560, 3061800, 3752892, 4193280, 5467500, 7500000, 8168160, 10022520, 11166960, 17203200, 17915040, 18462300, 21086208, 22394880, 28964040
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OFFSET
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1,1
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COMMENTS
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The area of the triangle (a,b,c) are given by Heron's formula, A = sqrt(s(s-a)(s-b)(s-c)), where the side lengths are a, b, c and semiperimeter s = (a+b+c)/2.
The areas of the nonprimitive triangles of sides (a*k^3, b*k^3, c*k^3), k = 1,2,... are in the sequence with the value A*k^6.
There may be multiple triangles with the same area (see the table of examples below).
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LINKS
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EXAMPLE
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The following table gives the initial values of (A, a, b, c):
+--------+------+-------+-------+
| A | a | b | c |
+--------+------+-------+-------+
| 480 | 8 | 123 | 125 |
| 4200 | 70 | 125 | 125 |
| 4200 | 125 | 125 | 240 |
| 5148 | 88 | 125 | 125 |
| 5148 | 125 | 125 | 234 |
| 7500 | 125 | 125 | 150 |
| 7500 | 125 | 125 | 200 |
| 30720 | 64 | 984 | 1000 |
| 65520 | 125 | 2088 | 2197 |
| 268800 | 560 | 1000 | 1000 |
| 268800 | 1000 | 1000 | 1920 |
| 329472 | 704 | 1000 | 1000 |
| 329472 | 1000 | 1000 | 1872 |
| 349920 | 216 | 3321 | 3375 |
.................................
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MATHEMATICA
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nn=600; lst={}; Do[s=(a^3+b^3+c)/2; If[IntegerQ[s], area2=s (s-a^3)(s-b^3) (s-c); If[0<area2&&IntegerQ[Sqrt[area2]], AppendTo[lst, Sqrt[area2]]]], {a, nn}, {b, a}, {c, 1, 50000}]; 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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