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