Revision History for A329536
(Underlined text is an addition;
strikethrough text is a deletion.)
Showing entries 1-10
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#11 by N. J. A. Sloane at Sat Dec 07 00:38:58 EST 2019
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#10 by N. J. A. Sloane at Sat Dec 07 00:38:36 EST 2019
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| NAME
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Integer areas of integer-sided triangles where the lengths of two of the sides are of cube lengthcubes.
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| COMMENTS
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The areasarea of the trianglestriangle (a,b,c) are given by Heron's formula, A = sqrt(s(s-a)(s-b)(s-c)), where itsthe side lengths are a, b, c and semiperimeter s = (a+b+c)/2.
There couldmay be multiple triangles sharingwith the same area (see the example table of examples below).
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| EXAMPLE
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The following table gives the firstinitial values of (A, a, b, c):
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proposed
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Discussion
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| Sat Dec 07
| 00:38
| N. J. A. Sloane: edited
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#9 by Giovanni Resta at Tue Nov 19 07:38:12 EST 2019
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Discussion
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| Tue Nov 19
| 08:01
| Michel Lagneau: OK Giovanni.
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#8 by Giovanni Resta at Tue Nov 19 07:37:20 EST 2019
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| COMMENTS
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For a givenThere area, thecould numberbe ofmultiple triangles sharing isthe notsame uniquearea (see the example table below).
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Discussion
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| Tue Nov 19
| 07:38
| Giovanni Resta: I thought that ""the number of triangles is not unique" did not sound right. I don't know if I've improved it...
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#7 by Jon E. Schoenfield at Sat Nov 16 05:12:58 EST 2019
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#6 by Jon E. Schoenfield at Sat Nov 16 05:12:55 EST 2019
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| COMMENTS
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For a samegiven area, the number of triangles is not unique (see the example table below).
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#5 by Michel Marcus at Sat Nov 16 01:15:48 EST 2019
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#4 by Michel Marcus at Sat Nov 16 01:15:35 EST 2019
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| COMMENTS
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For a same area, the number of triangles is not unique (see the example table below).
The following table gives the first values (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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| EXAMPLE
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The following table gives the first values (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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#3 by Michel Lagneau at Sat Nov 16 00:54:55 EST 2019
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#2 by Michel Lagneau at Sat Nov 16 00:53:47 EST 2019
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| NAME
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allocatedInteger areas of integer-sided triangles where two sides are forof Michelcube Lagneaulength.
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| DATA
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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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Subset of A188158.
The areas 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 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.
For a same area, the number of triangles is not unique (see the table below).
The following table gives the first values (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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Cf. A188158, A232461.
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| KEYWORD
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allocated
nonn
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| AUTHOR
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Michel Lagneau, Nov 16 2019
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| STATUS
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approved
editing
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