OFFSET
1,1
COMMENTS
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 given area, the number of triangles is not unique (see the example table below).
EXAMPLE
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 |
.................................
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
KEYWORD
nonn
AUTHOR
Michel Lagneau, Nov 16 2019
STATUS
proposed