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 A210207 Area A of the non-right triangles such that A, the sides, and the circumradius are integers. 3
 168, 432, 480, 624, 672, 768, 1320, 1512, 1536, 1560, 1680, 1728, 1848, 1920, 2040, 2304, 2376, 2496, 2520, 2688, 2856, 3024, 3072, 3240, 3696, 3720, 3840, 3864, 3888, 4104, 4200, 4320, 4536, 5280, 5376, 5616, 5712, 6000, 6048, 6144, 6240, 6552, 6720, 6912 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS A103251 gives the areas of right triangles with the same property (the area, the sides, and the circumradius are integers). Thus the intersection of this sequence with A103251 will give the areas of 2 families of triangles with the same property: one family of right triangles and one family of non-right triangles. For example a(3) = A103251(8) = 480 generates two triangles whose sides are (a,b,c) = (32, 50, 78) = > A = 480, R = 65, and 32^2 + 50^2 is no square; (a,b,c) = (20, 48, 52) = > A = 480, R = 26, and 20^2 + 48^2  = 52^2 is square. {a(n) intersection A103251} = {480, 1320, 1536, 1920, 2520, 3024, 3696, 3840, ...} LINKS Mohammad K. Azarian, Solution of problem 125: Circumradius and Inradius, Math Horizons, Vol. 16, No. 2 (Nov. 2008), p. 32. Eric W. Weisstein, MathWorld: Circumradius FORMULA Area A = sqrt(s*(s-a)*(s-b)*(s-c)) with s = (a+b+c)/2 (Heron's formula); Circumradius R = a*b*c/4A. EXAMPLE 168 is in the sequence because, for (a,b,c) = (14,30,40), A = sqrt(42*(42-14)*(42-30)*(42-40)) = 168, and 14^2 + 30^2 is no square. MAPLE T:=array(1..4000):nn:=400:k:=0:for a from 1 to nn do: for b from a to nn do: for c from b to nn do:  p:=(a+b+c)/2 : x:=p*(p-a)*(p-b)*(p-c): u:=a^2+b^2:if  x>0 then x1:=sqrt(x) : y:=a*b*c/(4*x1): else fi:if x1=floor(x1) and y = floor(y) and u <> c^2 then k:=k+1:T[k]:=x1:else fi:od:od:od: L := [seq(T[i], i=1..k)]:L1:=convert(T, set):A:=sort(L1, `<`): print(A): MATHEMATICA nn=400; lst={}; Do[s=(a+b+c)/2; If[IntegerQ[s], area2=s (s-a) (s-b) (s-c); If[0 < area2 && a^2 != b^2+c^2 && IntegerQ[Sqrt[area2]] && IntegerQ[a*b*c/(4*Sqrt[area2])], AppendTo[lst, Sqrt[area2]]]], {a, nn}, {b, a}, {c, b}]; Union[lst] CROSSREFS Cf. A103251, A208984, A188158. Sequence in context: A105915 A158219 A273771 * A303082 A247721 A342427 Adjacent sequences:  A210204 A210205 A210206 * A210208 A210209 A210210 KEYWORD nonn AUTHOR Michel Lagneau, Mar 18 2012 STATUS approved

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Last modified July 28 17:15 EDT 2021. Contains 346335 sequences. (Running on oeis4.)