A278051 - OEIS

%I #5 Nov 23 2016 09:47:06

%S 2,3,10,35,252,2772,6435,858,680680,12932920,5290740,121687020,

%T 1029659400,3088978200,582272390700,18050444111700,128701918800,

%U 25740383760,70301729698200,10043104242600,109530094869795600,523310453266801200,51193413906534900,481218090721428060

%N 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 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 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 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( 1/(t1[i]+t1[i+1]), i=1..nops(t1)-1);

%p ans:=[op(ans),t2];

%p od:

%p ans;

%p map(numer,ans); # A278050

%p map(denom,ans); # A278051

%Y Cf. A006843, A005728, A240877, A278046, A278050.

%K nonn,frac

%O 1,1

%A _N. J. A. Sloane_, Nov 23 2016