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A208984 Areas A of the triangles such that A, the sides, the circumradius and the inradius are integers. 10
24, 96, 120, 168, 216, 240, 336, 384, 432, 480, 600, 624, 672, 720, 768, 840, 864, 960, 1080, 1176, 1320, 1344, 1512, 1536, 1560, 1680, 1728, 1848, 1920, 1944, 2016, 2040, 2160, 2184, 2304, 2376, 2400, 2496, 2520, 2688, 2856, 2880, 2904, 3000, 3024, 3072, 3240 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

a(n) is divisible by 24, and the positive squares A000290(n) are included in the sequence a(n)/24 = {1, 4, 5, 7, 9, 10, 14, 16, 18, 20, 25, 26, 28, 30, 32, 35, 36, 40, 45, 49, 55, 56, 63, 64, 65, ...}.

The area A of a triangle whose sides have lengths a, b, and c is given by Heron's formula: A = sqrt(s*(s-a)*(s-b)*(s-c)), where s = (a+b+c)/2. The inradius r is given by r = A/s and the circumradius is given by R = abc/4A.

LINKS

Table of n, a(n) for n=1..47.

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

Eric W. Weisstein, MathWorld: Inradius

EXAMPLE

a(1) = 24 because, for (a,b,c) = (6, 8, 10) => s = (6 + 8 + 10)/2 = 12, and

A = sqrt(12(12-6)(12-8)(12-10)) = sqrt(576) = 24;

R = abc/4A = 480/4*24 = 5;

r = A/p = 24/12 = 2.

MAPLE

with(numtheory):T:=array(1..1000):k:=0:nn:=250: 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): if x>0 then s:=sqrt(x) :if s=floor(s) and irem(a*b*c, 4*s) = 0 and irem(s, p)=0 then k:=k+1:T[k]:= s: else fi:fi:od:od:od: L := [seq(T[i], i=1..k)]:L1:=convert(T, set):A:=sort(L1, `<`): print(A):

MATHEMATICA

nn = 1000; 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[a*b*c/(4* Sqrt[area2])] && IntegerQ[Sqrt[area2]/s], AppendTo[lst, Sqrt[area2]]]], {a, nn}, {b, a}, {c, b}]; Union[lst]

CROSSREFS

Cf. A120062, A188158.

Sequence in context: A055671 A090214 A283446 * A103251 A256418 A198387

Adjacent sequences:  A208981 A208982 A208983 * A208985 A208986 A208987

KEYWORD

nonn

AUTHOR

Michel Lagneau, Mar 04 2012

STATUS

approved

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Last modified July 22 07:23 EDT 2019. Contains 325216 sequences. (Running on oeis4.)