MathDB

Let

n > 1

be a positive integer. List in increasing order all the irreducible fractions in the interval

[0, 1]

that have a positive denominator less than or equal to

n

\frac{0}{1} = \frac{p_0}{q_0} < \frac{p_1}{q_1} < \cdots < \frac{p_M}{q_M} = \frac{1}{1}.

Determine, in function of

n

, the smallest possible value of

q_{i-1} + q_i + q_{i+1}

, for

0 < i < M

.
For example, if

n = 4

, the enumeration is

\frac{0}{1} < \frac{1}{4} < \frac{1}{3} < \frac{1}{2} < \frac{2}{3} < \frac{3}{4} < \frac{1}{1},

where

p_0 = 0, p_1 = 1, p_2 = 1, p_3 = 1, p_4 = 2, p_5 = 3, p_6 = 1, q_0 = 1, q_1 = 4, q_2 = 3, q_3 = 2, q_4 = 3, q_5 = 4, q_6 = 1

, and the minimum is

1 + 4 + 3 = 3 + 2 + 3 = 3 + 4 + 1 = 8

6