MathDB

Let

k

be a positive integer. A sequence

a_0,a_1,...,a_n,n>0

of positive integers satisfies the following conditions:

(i)

a_0=a_n=1

;

(ii)

2\leq a_i\leq k

for each

i=1,2,...,n-1

;

(iii)

For each

j=2,3,...,k

, the number

j

appears

\phi(j)

times in the sequence

a_0,a_1,...,a_n

, where

\phi(j)

is the number of positive integers that do not exceed

j

and are coprime to

j

;

(iv)

For any

i=1,2,...,n-1

\gcd(a_i,a_{i-1})=1=\gcd(a_i,a_{i+1})

, and

a_i

divides

a_{i-1}+a_{i+1}

. Suppose there is another sequence

b_0,b_1,...,b_n

of integers such that

\frac{b_{i+1}}{a_{i+1}}>\frac{b_i}{a_i}

for all

i=0,1,...,n-1

. Find the minimum value of

b_n-b_0

Problem Statement