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A343893 Side c of integer-sided primitive triangles (a, b, c) where side a is the harmonic mean of the 2 other sides b and c, i.e., 2/a = 1/b + 1/c with b < a < c.

%I A343893 #11 Apr 03 2023 14:36:11
%S A343893 6,15,20,35,28,42,45,63,88,77,66,72,117,99,104,91,130,110,165,120,143,
%T A343893 204,187,170,156,153,221,247,195,228,266,209,190,238,210,273,285,231,
%U A343893 255,368,336,345,304,322,391,272,299,276,425,357,450,323,400,414,513,350,325,342,475,459
%N A343893 Side c of integer-sided primitive triangles (a, b, c) where side a is the harmonic mean of the 2 other sides b and c, i.e., 2/a = 1/b + 1/c with b < a < c.
%C A343893 The triples (a, b, c) are displayed in increasing order of side a, and if sides a coincide then in increasing order of the side b.
%C A343893 The sequence is not increasing because a(4) = 35 > a(5) =  28, but, these sides c are listed in increasing order in A020886.
%C A343893 For the corresponding primitive triples and miscellaneous properties and references, see A343891.
%F A343893 a(n) = A343891(n, 3).
%e A343893 a(3) = 20, because the third triple is (15, 12, 20) with side c = 20, satisfying 1/20 = 2/15 - 1/12 and 15-12 < 20 < 15+12.
%p A343893 for a from 4 to 200 do
%p A343893 for b from floor(a/2)+1 to a-1 do
%p A343893 c := a*b/(2*b-a);
%p A343893 if c=floor(c) and igcd(a,b,c)=1 and c-b<a then print(c); end if;
%p A343893 end do;
%p A343893 end do;
%Y A343893 Cf. A343891 (triples), A020883 (side a), A343892 (side b), A343894 (perimeter).
%Y A343893 Cf. A020886 (sides c ordered).
%K A343893 nonn
%O A343893 1,1
%A A343893 _Bernard Schott_, May 06 2021