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A343892 Side b 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 A343892 #12 Apr 03 2023 14:35:09
%S A343892 3,10,12,15,21,30,36,35,40,44,55,56,52,63,65,78,70,90,77,105,99,85,
%T A343892 102,119,132,136,117,114,143,133,126,152,171,154,182,168,165,210,195,
%U A343892 161,176,184,208,207,187,240,230,253,200,221,198,255,225,234,216,275,300,306,247,270
%N A343892 Side b 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 A343892 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 A343892 The sequence is not increasing because a(7) = 36 > a(8) = 35, but, these sides b are listed in increasing order in A020890.
%C A343892 The first term appearing twice is 330 and corresponds to triples (435, 330, 638) and (460, 330, 759), the second one is 462 and corresponds to triples (483, 462, 506) and (532, 462, 627).
%C A343892 For the corresponding primitive triples and miscellaneous properties and references, see A343891.
%F A343892 a(n) = A343891(n, 2).
%e A343892 a(4) = 15, because the fourth triple is (21, 15, 35) with side b = 15, satisfying 1/15 = 2/21 - 1/35 and 31-15 < 21 < 31+15.
%p A343892 for a from 4 to 200 do
%p A343892 for b from floor(a/2)+1 to a-1 do
%p A343892 c := a*b/(2*b-a);
%p A343892 if c=floor(c) and igcd(a,b,c)=1 and c-b<a then print(b); end if;
%p A343892 end do;
%p A343892 end do;
%Y A343892 Cf. A343891 (triples), A020883 (side a), A343893 (side c), A343894 (perimeter).
%Y A343892 Cf. A020890 (sides b ordered).
%K A343892 nonn
%O A343892 1,1
%A A343892 _Bernard Schott_, May 06 2021