A329536 Integer areas of integer-sided triangles where the lengths of two of the sides are cubes.

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

Offset

1,1

Comments

Subset of A188158.

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).

Links

Table of n, a(n) for n=1..31.

Example

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 |
.................................

Mathematica

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]

Crossrefs

Cf. A188158, A232461.

Sequence in context: A327944 A357493 A063870 * A331768 A035314 A022047

Adjacent sequences: A329533 A329534 A329535 * A329537 A329538 A329539

Keyword

nonn

Author

Michel Lagneau, Nov 16 2019

Status

approved