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Integer inradii for which there exists an isosceles triangle with integer sides (a, a, c) where a < c.
3

%I #22 Jun 27 2023 05:15:57

%S 3,4,6,8,9,12,15,16,18,20,21,24,27,28,30,32,33,35,36,39,40,42,44,45,

%T 48,51,52,54,56,57,60,63,64,66,68,69,70,72,75,76,78,80,81,84,87,88,90,

%U 92,93,96,99,100,102,104,105,108,111,112,114,116,117,120,123,124,126,128,129,132,135

%N Integer inradii for which there exists an isosceles triangle with integer sides (a, a, c) where a < c.

%C The inradius for isosceles triangle (a, a, c) is r = (c/2)*sqrt((2*a-c)/(2*a+c)).

%C If m is a term, so is k*m with k > 1.

%C As r = 3 and r = 4 are terms, A008585 and A008586 are respective subsequences; the only terms < 100 that are not multiples of 3 or 4 are 35 and 70, the next one is r = 154 = 2*7*11 for triple (765, 765, 1386).

%C By the triangle inequality, a+1 <= c <= 2*a-1.

%C Differs from A059267. Examples: 154 is not in A059267 but in this sequence at radius r=154 with side lengths c=1386 and a=765. 442 is not in A059267 but in this sequences with r=442, c=6630, a=3435. - _R. J. Mathar_, Jun 26 2023

%H R. J. Mathar, <a href="/A362670/a362670_1.pdf">Solution strategy and Maple program</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Incircle.html">Incircle</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IsoscelesTriangle.html">Isosceles Triangle</a>.

%e The smallest inradius r = 3 corresponds to isosceles triangle (10, 10, 12).

%e The second inradius r = 4 corresponds to isosceles triangle (15, 15, 24).

%e r = 15 is the first inradius for which there exist two such isosceles triangles: (50, 50, 60) and (68, 68, 120).

%e r = 35 is the smallest inradius that is not multiple of 3 or of 4, this inradius corresponds to isosceles triangle (222, 222, 420).

%Y Cf. A008585, A008586, A070204, A120062, A120570.

%Y Cf. A362669 (similar but with (a,b,b)).

%K nonn

%O 1,1

%A _Bernard Schott_, May 05 2023