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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.

%I A247381 #13 Jun 15 2016 12:06:55
%S A247381 36,72,12,36,180,6,252,72,36,90,396,12,468,126,60,7056,2448,72,684,
%T A247381 180,84,198,20700,24,900,234,5292,252,4176,30,1116,288,132,306,1260,
%U A247381 36,1332,5472,156,360,5904,42,1548,396,180,1656,82908,1200,7056,1800,204,468,30528,216
%N 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.
%C A247381 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.
%H A247381 N. J. Fine, <a href="http://www.jstor.org/stable/2319348?seq=1#page_scan_tab_contents">On Rational Triangles</a>, Mathematical Association of America,  83-7 (1976), 517-521.
%H A247381 Jaap Top and Noriko Yui, <a href="http://www.msri.org/people/staff/levy/files/Book44/19yui.pdf">Congruent number problems and their variants</a>, Algorithmic Number Theory, MSRI Publications Volume 44, 2008, p. 621.
%H A247381 Yahoo Answers, <a href="https://answers.yahoo.com/question/index?qid=20111217134327AAK0rKW">Is each positive integer the area of some triangle with rational sides?</a>
%e A247381 a(23)=30^2*23=20700 and the primitive Heronian triangle has sides (73, 579, 598).
%t A247381 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}]
%Y A247381 Cf. A224301.
%K A247381 nonn
%O A247381 1,1
%A A247381 _Frank M Jackson_, Sep 15 2014
%E A247381 Updated and edited by _Frank M Jackson_, Jun 14 2016