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A362670 Integer inradii for which there exists an isosceles triangle with integer sides (a, a, c) where a < c.

%I A362670 #23 Feb 16 2025 08:34:05
%S A362670 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 A362670 48,51,52,54,56,57,60,63,64,66,68,69,70,72,75,76,78,80,81,84,87,88,90,
%U A362670 92,93,96,99,100,102,104,105,108,111,112,114,116,117,120,123,124,126,128,129,132,135
%N A362670 Integer inradii for which there exists an isosceles triangle with integer sides (a, a, c) where a < c.
%C A362670 The inradius for isosceles triangle (a, a, c) is r = (c/2)*sqrt((2*a-c)/(2*a+c)).
%C A362670 If m is a term, so is k*m with k > 1.
%C A362670 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 A362670 By the triangle inequality, a+1 <= c <= 2*a-1.
%C A362670 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 A362670 R. J. Mathar, <a href="/A362670/a362670_1.pdf">Solution strategy and Maple program</a>
%H A362670 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Incircle.html">Incircle</a>.
%H A362670 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IsoscelesTriangle.html">Isosceles Triangle</a>.
%e A362670 The smallest inradius r = 3 corresponds to isosceles triangle (10, 10, 12).
%e A362670 The second inradius r = 4 corresponds to isosceles triangle (15, 15, 24).
%e A362670 r = 15 is the first inradius for which there exist two such isosceles triangles: (50, 50, 60) and (68, 68, 120).
%e A362670 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 A362670 Cf. A008585, A008586, A070204, A120062, A120570.
%Y A362670 Cf. A362669 (similar but with (a,b,b)).
%K A362670 nonn
%O A362670 1,1
%A A362670 _Bernard Schott_, May 05 2023