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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 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

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Last modified September 18 21:47 EDT 2024. Contains 376002 sequences. (Running on oeis4.)