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).

1, 1, 7, 4, 37, 53, 707, 85, 179077, 289613, 379721, 641671, 62836087, 35819033, 6367281023, 55181728027, 13442946373, 490167893, 596530310479, 576997238399, 116144361532321, 4931206160615, 164890340129357, 1514840590670747, 10181612956306486603, 3295813969039399097

Offset

1,3

Links

Table of n, a(n) for n=1..26.

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

Example

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, ...

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 50 do
t1:=denom(Farey(n));
t2:=add( 1/(t1[i]*t1[i+1]*(t1[i]+t1[i+1])), i=1..nops(t1)-1);
ans:=[op(ans), t2];
od:
ans;
map(numer, ans); # A278047
map(denom, ans); # A278048

Crossrefs

Cf. A006843, A005728, A240877, A278046, A278048.

Sequence in context: A270732 A270468 A270983 * A271007 A271053 A270629

Adjacent sequences: A278044 A278045 A278046 * A278048 A278049 A278050

Keyword

nonn,frac

Author

N. J. A. Sloane, Nov 22 2016

Status

approved