

A247381


The area of a primitive Heronian triangle K, such that K = k^2*n for the least k, where n is the sequence index.


1



36, 72, 12, 36, 180, 6, 252, 72, 36, 90, 396, 12, 468, 126, 60, 7056, 2448, 72, 684, 180, 84, 198, 20700, 24, 900, 234, 5292, 252, 4176, 30, 1116, 288, 132, 306, 1260, 36, 1332, 5472, 156, 360, 5904, 42, 1548, 396, 180, 1656, 82908, 1200, 7056, 1800, 204, 468, 30528, 216
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

It has been proved that every positive integer is the area of some rational sided Heronian triangle. Therefore for all positive integers n there exists a primitive Heronian triangle such that for some least k^2 its area K = k^2*n. The Mathematica program limits searches to all primitive Heronian triangles whose largest side does not exceed 1000 and returns 0 if no area is found.


LINKS



EXAMPLE

a(23)=30^2*23=20700 and the primitive Heronian triangle has sides (73, 579, 598).


MATHEMATICA

getarea[n0_] := (area1=0; Do[If[IntegerQ[area=Sqrt[(a+b+c)(a+bc)(ab+c)(a+b+c)/16]]&&area>0&&IntegerQ[k=Sqrt[area/n0]]&&GCD[a, b, c]==1, area1=area; Break[]], {c, 3, 1000}, {b, 1, c}, {a, 1, b}]; area1); Table[getarea[n], {n, 1, 100}]


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



