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 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 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. 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

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Last modified July 4 02:21 EDT 2020. Contains 335436 sequences. (Running on oeis4.)