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A278561 Let v = list of denominators of Farey series of order n (see A006843); let b(n) = Sum k*k'/(k+k'), where (k,k') are pairs of successive terms of v; a(n) = denominator of b(n).

%I A278561 #7 Nov 23 2016 09:56:54
%S A278561 2,3,10,7,252,396,6435,858,680680,1175720,5290740,9360540,1029659400,
%T A278561 617795640,116454478140,1061790830100,283144221360,74511637200,
%U A278561 14060345939640,14060345939640,109530094869795600,650075097225840,51193413906534900,481218090721428060
%N A278561 Let v = list of denominators of Farey series of order n (see A006843); let b(n) = Sum k*k'/(k+k'), where (k,k') are pairs of successive terms of v; a(n) = denominator of b(n).
%H A278561 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. (21).
%e A278561 The fractions b(n) are 1/2, 4/3, 39/10, 52/7, 4069/252, 8573/396, 258017/6435, 46639/858, 53371999/680680, 113518551/1175720, 768140741/5290740, 1560819091/9360540, 242830653007/1029659400, 169134016817/617795640, 38186305937387/116454478140, ...
%p A278561 Farey := proc(n) sort(convert(`union`({0}, {seq(seq(m/k, m=1..k), k=1..n)}), list)) end:
%p A278561 ans:=[];
%p A278561 for n from 1 to 30 do
%p A278561 t1:=denom(Farey(n));
%p A278561 t2:=add( t1[i]*t1[i+1]/(t1[i]+t1[i+1]), i=1..nops(t1)-1);
%p A278561 od:
%p A278561 ans;
%p A278561 map(numer,ans); # A278052
%p A278561 map(denom,ans); # A278561
%Y A278561 Cf. A006843, A005728, A240877, A278046-A278051, A278052 (numerators).
%K A278561 nonn,frac
%O A278561 1,1
%A A278561 _N. J. A. Sloane_, Nov 23 2016