Voozh

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

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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.

FORMULA

a(n) = (floor(sqrt(n)) + floor(sqrt(n-1)) + 2*floor(sqrt(n)-1/2) - tau(n))/2. - Ridouane Oudra, Feb 20 2026

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,