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

1,1

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

Table of n, a(n) for n=1..54.

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?

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+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}]

Crossrefs

Cf. A224301.

Sequence in context: A182428 A208128 A260919 * A249726 A335463 A192026

Adjacent sequences: A247378 A247379 A247380 * A247382 A247383 A247384

Keyword

nonn

Author

Frank M Jackson, Sep 15 2014

Extensions

Updated and edited by Frank M Jackson, Jun 14 2016

Status

approved