Question 97
Let $k$ be a positive integer. A sequence $a_0, a_1, \dots, a_n$ ($n \gt 0$) of positive integers satisfies the following conditions:
(i) $a_0 = a_n = 1$;
(ii) $2 \le a_i \le k$ for each $i = 1, 2, \dots, n-1$;
(iii) For each $j = 2, 3, \dots, k$, the number $j$ appears $\varphi(j)$ times in the sequence $a_0, a_1, \dots, a_n$ ($\varphi(j)$ is the number of positive integers that do not exceed $j$ and are coprime to $j$);
(iv) For any $i = 1, 2, \dots, n-1$, $\gcd(a_{i-1}, a_i) = 1 = \gcd(a_i, a_{i+1})$, and $a_i$ divides $a_{i-1} + a_{i+1}$.
There is another sequence $b_0, b_1, \dots, b_n$ of integers such that $\frac{b_{i+1}}{a_{i+1}} \gt \frac{b_i}{a_i}$ for all $i = 0, 1, \dots, n-1$. Find the minimum value for $b_n - b_0$.
(i) $a_0 = a_n = 1$;
(ii) $2 \le a_i \le k$ for each $i = 1, 2, \dots, n-1$;
(iii) For each $j = 2, 3, \dots, k$, the number $j$ appears $\varphi(j)$ times in the sequence $a_0, a_1, \dots, a_n$ ($\varphi(j)$ is the number of positive integers that do not exceed $j$ and are coprime to $j$);
(iv) For any $i = 1, 2, \dots, n-1$, $\gcd(a_{i-1}, a_i) = 1 = \gcd(a_i, a_{i+1})$, and $a_i$ divides $a_{i-1} + a_{i+1}$.
There is another sequence $b_0, b_1, \dots, b_n$ of integers such that $\frac{b_{i+1}}{a_{i+1}} \gt \frac{b_i}{a_i}$ for all $i = 0, 1, \dots, n-1$. Find the minimum value for $b_n - b_0$.