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%I #6 Nov 23 2016 09:55:15
%S 1,4,39,52,4069,8573,258017,46639,53371999,113518551,768140741,
%T 1560819091,242830653007,169134016817,38186305937387,408881289764107,
%U 143220706672837,41293923006131,9928250098118791,10936700271572951,97615258031147892517,643700119549549507,62211198375587838727
%N 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) = numerator of b(n).
%H 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 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 Farey := proc(n) sort(convert(`union`({0}, {seq(seq(m/k, m=1..k), k=1..n)}), list)) end:
%p ans:=[];
%p for n from 1 to 30 do
%p t1:=denom(Farey(n));
%p t2:=add( t1[i]*t1[i+1]/(t1[i]+t1[i+1]), i=1..nops(t1)-1);
%p od:
%p ans;
%p map(numer,ans); # A278052
%p map(denom,ans); # A278561
%Y Cf. A006843, A005728, A240877, A278046-A278051, A278561 (denominators).
%K nonn,frac
%O 1,2
%A _N. J. A. Sloane_, Nov 23 2016
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