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A120063 Shortest side c of all integer-sided triangles with sides a<=b<=c and inradius n.

%I A120063 #12 Jan 28 2023 10:36:10
%S A120063 5,10,12,15,25,24,35,30,36,39,55,45,65,63,53,60,85,68,95,75,77,88,115,
%T A120063 85,125,130,108,105,145,106,155,120,132,170,137,135,185,190,156,150,
%U A120063 205,154,215,165,159,230,235,170,245,195,204,195,265,204,200,195,228,290
%N A120063 Shortest side c of all integer-sided triangles with sides a<=b<=c and inradius n.
%C A120063 Terms a(11),..., a(100) computed by Thomas Mautsch (mautsch(AT)ethz.ch).
%C A120063 Empirically, 2*sqrt(3) < a(n)/n <= 5. The lower bound is provably tight, the upper bound seems to be achieved infinitely often, e.g, for prime n >= 5. It appears that a(p) = 5p for prime p != 3. - _David W. Wilson_, Jun 17 2006
%C A120063 Minimum of longest side occurring among all A120062(n) triangles having integer sides with integer inradius n.
%D A120063  Mohammad K. Azarian, Circumradius and Inradius, Problem S125, Math Horizons, Vol. 15, Issue 4, April 2008, p. 32.  Solution published in Vol. 16, Issue 2, November 2008, p. 32.
%H A120063 David W. Wilson, <a href="/A120063/b120063.txt">Table of n, a(n) for n = 1..10000</a>
%e A120063 a(1)=5 because the only triangle with integer sides and inradius 1 is {3,4,5}; its longest side is 5.
%e A120063 a(2)=10: The triangles with inradius 2 are {5,12,13}, {6,8,10}, {6,25,29}, {7,15,20}, {9,10,17}. The minimum of their longest sides is min(13,10,29,20,17)=10.
%Y A120063 See A120062 for sequences related to integer-sided triangles with integer inradius n.
%Y A120063 Cf. A120062 [triangles with integer inradius], A120252 [primitive triangles with integer inradius], A057721 [maximum of longest sides], A058331 [maximum of shortest sides], A120064 [minimum of middle sides], A082044 [maximum of middle sides], A005408 [minimum of shortest sides], A007237.
%K A120063 nonn
%O A120063 1,1
%A A120063 _Hugo Pfoertner_, Jun 13 2006