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A278051 Let v = list of denominators of Farey series of order n (see A006843); let b(n) = Sum 1/(k+k'), where (k,k') are pairs of successive terms of v; a(n) = denominator of b(n).

%I A278051 #5 Nov 23 2016 09:47:06
%S A278051 2,3,10,35,252,2772,6435,858,680680,12932920,5290740,121687020,
%T A278051 1029659400,3088978200,582272390700,18050444111700,128701918800,
%U A278051 25740383760,70301729698200,10043104242600,109530094869795600,523310453266801200,51193413906534900,481218090721428060
%N A278051 Let v = list of denominators of Farey series of order n (see A006843); let b(n) = Sum 1/(k+k'), where (k,k') are pairs of successive terms of v; a(n) = denominator of b(n).
%H A278051 J. Lehner and M. Newman, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa15/aa15114.pdf">Sums involving Farey fractions</a>, Acta Arithmetica 15.2 (1969): 181-187. See Eq. (20).
%e A278051 The fractions b(n) are 1/2, 2/3, 9/10, 38/35, 347/252, 4189/2772, 11767/6435, 1733/858, 1548081/680680, 31464371/12932920, 14680543/5290740, 353517989/121687020, 3350216417/1029659400, 10571768267/3088978200, ...
%p A278051 Farey := proc(n) sort(convert(`union`({0}, {seq(seq(m/k, m=1..k), k=1..n)}), list)) end:
%p A278051 ans:=[];
%p A278051 for n from 1 to 30 do
%p A278051 t1:=denom(Farey(n));
%p A278051 t2:=add( 1/(t1[i]+t1[i+1]), i=1..nops(t1)-1);
%p A278051 ans:=[op(ans),t2];
%p A278051 od:
%p A278051 ans;
%p A278051 map(numer,ans); # A278050
%p A278051 map(denom,ans); # A278051
%Y A278051 Cf. A006843, A005728, A240877, A278046, A278050.
%K A278051 nonn,frac
%O A278051 1,1
%A A278051 _N. J. A. Sloane_, Nov 23 2016