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Perimeters of integer-sided triangles such that the harmonic mean of all the side lengths and the harmonic mean of each pair of side lengths is an integer.
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%I #6 Apr 20 2023 18:21:05

%S 3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60,63,66,69,70,

%T 72,75,78,81,84,87,90,93,96,99,102,105,108,111,114,117,120,123,126,

%U 129,132,135,138,140,141,144,147,150,153,156,159,162,165,168,171,174,177,180,183,186

%N Perimeters of integer-sided triangles such that the harmonic mean of all the side lengths and the harmonic mean of each pair of side lengths is an integer.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Integer_triangle">Integer Triangle</a>

%e 6 is in the sequence since the integer-sided triangle [2,2,2] (with perimeter 6) has harmonic mean 3*2*2*2/(2*2+2*2+2*2) = 2 (an integer), and the harmonic mean of each pair of side lengths has harmonic mean 2*2*2/(2+2) = 2 (an integer).

%e 14 is not in the sequence. Although each pair of side lengths in the integer triangle [2,6,6] (with perimeter 14) has an integer harmonic mean (i.e., 2*2*6/(2+6) = 3, 2*2*6/(2+6) = 3 and 2*6*6/(6+6) = 72/12 = 6), the harmonic mean of all the side lengths is 3*2*6*6/(2*6+2*6+6*6) = 216/60 = 18/5 (not an integer).

%Y Cf. A337088, A337091.

%K nonn

%O 1,1

%A _Wesley Ivan Hurt_, Aug 20 2020