%N a(n) is the number of open intervals (m, m+1) containing at least one fraction n/k, where m and k are integers between 1 and n.
%C a(n) is the number of distinct values floor(n/k) can take, for k an integer between 1 and n that does not divide n.
%e Fractions with 9 as the numerator, with a denominator not greater than the numerator, and that do not yield an integer, are:
%e 9/2 that lies between 4 and 5,
%e 9/4 between 2 and 3,
%e 9/5, 9/6, 9/7, 9/8 between 1 and 2.
%e Three intervals are involved, so a(9)=3.
%t a[n_] := Length[ Union[Floor /@ Select[n/# & /@ Range[n], ! IntegerQ[#] &]]]; Array[a, 80]
%Y Cf. A055086 (when k is allowed to divide n).
%Y Cf. A316519 (when frac(n/k) is used instead of floor(n/k)).
%A _Luc Rousseau_, Jul 24 2018