cp's OEIS Frontend

A278050 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) = numerator of b(n).

%I A278050 #7 Nov 23 2016 09:45:55
%S A278050 1,2,9,38,347,4189,11767,1733,1548081,31464371,14680543,353517989,
%T A278050 3350216417,10571768267,2114915577977,69039991480573,538250871701,
%U A278050 110983833313,328448743696081,48484885139543,553270527392631611,2736415713954900433,286367762285513933,2754025786313797907
%N A278050 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) = numerator of b(n).
%H A278050 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 A278050 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 A278050 Farey := proc(n) sort(convert(`union`({0}, {seq(seq(m/k, m=1..k), k=1..n)}), list)) end:
%p A278050 ans:=[];
%p A278050 for n from 1 to 30 do
%p A278050 t1:=denom(Farey(n));
%p A278050 t2:=add( 1/(t1[i]+t1[i+1]), i=1..nops(t1)-1);
%p A278050 ans:=[op(ans),t2];
%p A278050 od:
%p A278050 ans;
%p A278050 map(numer,ans); # A278050
%p A278050 map(denom,ans); # A278051
%Y A278050 Cf. A006843, A005728, A240877, A278046, A278051.
%K A278050 nonn,frac
%O A278050 1,2
%A A278050 _N. J. A. Sloane_, Nov 23 2016