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).

1, 2, 9, 38, 347, 4189, 11767, 1733, 1548081, 31464371, 14680543, 353517989, 3350216417, 10571768267, 2114915577977, 69039991480573, 538250871701, 110983833313, 328448743696081, 48484885139543, 553270527392631611, 2736415713954900433, 286367762285513933, 2754025786313797907

Offset: 1

Author

N. J. A. Sloane, Nov 23 2016

Keywords

• nonn
• frac

Examples

			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, ...
		

Links

• J. Lehner and M. Newman, Sums involving Farey fractions, Acta Arithmetica 15.2 (1969): 181-187. See Eq. (20).

Crossrefs

Cf. A006843, A005728, A240877, A278046, A278051.

Programs

Maple

Farey := proc(n) sort(convert(`union`({0}, {seq(seq(m/k, m=1..k), k=1..n)}), list)) end:
ans:=[];
for n from 1 to 30 do
t1:=denom(Farey(n));
t2:=add( 1/(t1[i]+t1[i+1]), i=1..nops(t1)-1);
ans:=[op(ans),t2];
od:
ans;
map(numer,ans); # A278050
map(denom,ans); # A278051