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A278047 Let v = list of denominators of Farey series of order n (see A006843); let b(n) = Sum 1/(k*k'*(k+k')), where (k,k') are pairs of successive terms of v; a(n) = numerator of b(n).

%I A278047 #14 Nov 23 2016 02:49:24
%S A278047 1,1,7,4,37,53,707,85,179077,289613,379721,641671,62836087,35819033,
%T A278047 6367281023,55181728027,13442946373,490167893,596530310479,
%U A278047 576997238399,116144361532321,4931206160615,164890340129357,1514840590670747,10181612956306486603,3295813969039399097
%N A278047 Let v = list of denominators of Farey series of order n (see A006843); let b(n) = Sum 1/(k*k'*(k+k')), where (k,k') are pairs of successive terms of v; a(n) = numerator of b(n).
%H A278047 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.
%e A278047 The fractions b(n) are 1/2, 1/3, 7/30, 4/21, 37/252, 53/396, 707/6435, 85/858, 179077/2042040, 289613/3527160, 379721/5290740, 641671/9360540, 62836087/1029659400, 35819033/617795640, ...
%p A278047 Farey := proc(n) sort(convert(`union`({0}, {seq(seq(m/k, m=1..k), k=1..n)}), list)) end:
%p A278047 ans:=[];
%p A278047 for n from 1 to 50 do
%p A278047 t1:=denom(Farey(n));
%p A278047 t2:=add( 1/(t1[i]*t1[i+1]*(t1[i]+t1[i+1])),i=1..nops(t1)-1);
%p A278047 ans:=[op(ans),t2];
%p A278047 od:
%p A278047 ans;
%p A278047 map(numer,ans); # A278047
%p A278047 map(denom,ans); # A278048
%Y A278047 Cf. A006843, A005728, A240877, A278046, A278048.
%K A278047 nonn,frac
%O A278047 1,3
%A A278047 _N. J. A. Sloane_, Nov 22 2016