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.

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

Offset: 1

Author

Frank M Jackson, Sep 15 2014

Keywords

• nonn

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.

Examples

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

Links

• N. J. Fine, On Rational Triangles, Mathematical Association of America, 83-7 (1976), 517-521.
• Jaap Top and Noriko Yui, Congruent number problems and their variants, Algorithmic Number Theory, MSRI Publications Volume 44, 2008, p. 621.
• Yahoo Answers, Is each positive integer the area of some triangle with rational sides?

Crossrefs

Cf. A224301.

Programs

Mathematica

getarea[n0_] := (area1=0; Do[If[IntegerQ[area=Sqrt[(a+b+c)(a+b-c)(a-b+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}]

Extensions

Updated and edited by Frank M Jackson, Jun 14 2016