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, 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, 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, 12, 11, 11, 13, 11, 14, 12, 13, 11, 14, 9, 15, 14, 13, 13, 14, 12, 15, 11

Offset

1,5

Comments

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.

Links

Table of n, a(n) for n=1..80.

Example

Fractions with 9 as the numerator, with a denominator not greater than the numerator, and that do not yield an integer, are:
9/2 that lies between 4 and 5,
9/4 between 2 and 3,
9/5, 9/6, 9/7, 9/8 between 1 and 2.
Three intervals are involved, so a(9)=3.

Mathematica

a[n_] := Length[ Union[Floor /@ Select[n/# & /@ Range[n], ! IntegerQ[#] &]]]; Array[a, 80]

Crossrefs

Cf. A055086 (when k is allowed to divide n).

Cf. A316519 (when frac(n/k) is used instead of floor(n/k)).

Sequence in context: A253281 A029206 A029200 * A051274 A267806 A025797

Adjacent sequences: A317240 A317241 A317242 * A317244 A317245 A317246

Keyword

nonn

Author

Luc Rousseau, Jul 24 2018

Status

approved