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A317243 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. 0

%I

%S 0,0,1,1,2,1,3,2,3,3,4,2,5,4,4,4,6,4,6,4,6,6,7,4,7,7,7,6,8,5,9,7,8,8,

%T 8,6,10,9,9,7,10,7,11,9,9,10,11,7,11,10,11,10,12,9,11,9,12,12,13,8,13,

%U 12,11,11,13,11,14,12,13,11,14,9,15,14,13,13,14,12,15,11

%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)).

%K nonn

%O 1,5

%A _Luc Rousseau_, Jul 24 2018

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Last modified October 26 08:00 EDT 2021. Contains 348267 sequences. (Running on oeis4.)