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 A210643 Areas A of the triangles such that A, the sides and the three altitudes are integers. 6
 150, 300, 600, 1050, 1200, 1350, 2400, 2700, 3750, 4200, 4800, 5070, 5400, 7350, 7500, 9450, 9600, 10140, 10800, 12150, 14700, 15000, 16800, 17340, 18150, 19200, 20280, 21600, 24300, 25350, 26250, 29400, 30000, 33750, 34680, 36300, 37800 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Properties of this sequence : There exists three class of numbers included into a(n) : (i) A subset such that {150, 600, 1350, 2400, 3750, 5070,…} where the sides a h1 = b, h2 = a, h3 = a*b/c. (ii) A subset such that a(n) = 300*n^2 = {300, 1200, 2700, 4800, …} where the triangles (a,b,c) are isosceles with a = b < c, and it is easy to check that a = b = 25*n,  c=30*n, h1 = h2 = 24*n and h3 = sqrt(b^2 - c^2/4). (iii) A subset such that {1050, 4200, 9450,…} without the precedent properties. LINKS Ray Chandler, Table of n, a(n) for n = 1..64 Eric Weisstein, Isosceles Triangle Eric Weisstein, Altitude EXAMPLE Primitive solutions follow: Area,  ( a,   b,   c),  (h1,  h2,  h3), Case   150,  (15,  20,  25),  (20,  15,  12), Right,   300,  (25,  25,  30),  (24,  24,  20), Isosceles,   300,  (25,  25,  40),  (24,  24,  15), Isosceles, 1050,  (35,  75, 100),  (60,  28,  21), Other, 5070,  (65, 156, 169), (156,  65,  60), Right, 10140, (130, 169, 169), (156, 120, 120), Isosceles, 10140, (169, 169, 312), (120, 120,  65), Isosceles, 17340, (136, 255, 289), (255, 136, 120), Right, 34680, (272, 289, 289), (255, 240, 240), Isosceles, 34680, (289, 289, 510), (240, 240, 136), Isosceles, 52500, (175, 600, 625), (600, 175, 168), Right, 82500, (275, 625, 750), (600, 264, 220), Other. MAPLE with(numtheory):T:=array(1..1000):k:=0:nn:=500: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:s:=p*(p-a)*(p-b)*(p-c):if s>0 then s1:=sqrt(s): h1:=2*s1/a: h2:=2*s1/b:h3:=2*s1/c:if s1=floor(s1) and h1=floor(h1) and h2=floor(h2) and h3=floor(h3) then k:=k+1:T[k]:=s1:else fi:fi:od:od:od: L := [seq(T[i], i=1..k)]:L1:=convert(T, set):A:=sort(L1, `<`): print(A): MATHEMATICA nn = 900; lst = {}; Do[s = (a + b + c)/2; If[IntegerQ[s], area2 = s (s - a) (s - b) (s - c); If[0 < area2 <= nn^2 && IntegerQ[Sqrt[area2]]&&IntegerQ[(2*Sqrt[area2])/a] &&IntegerQ[(2*Sqrt[area2])/b] &&IntegerQ[(2*Sqrt[area2])/c], AppendTo[lst, Sqrt[area2]]]], {a, nn}, {b, a}, {c, b}]; Union[lst] CROSSREFS Cf. A188158. Sequence in context: A335145 A063829 A291959 * A211550 A212465 A273322 Adjacent sequences:  A210640 A210641 A210642 * A210644 A210645 A210646 KEYWORD nonn AUTHOR Michel Lagneau, Mar 26 2012 EXTENSIONS More terms from Ray Chandler, Apr 24 2013 STATUS approved

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Last modified June 22 03:54 EDT 2021. Contains 345367 sequences. (Running on oeis4.)