Let $f_a:\mathbb N\to \mathbb N$ with $f_a(1) = 1$, $f_a(2) = a$ for some $a\in \mathbb N$ and, for each positive integer $n\ge 3$, $f_a(n)$ is the smallest value not assumed at lower integers that is co-prime with $f_a(n-1)$. It is known that $f_a$ is onto. In fact these are permutations of $\mathbb N$ (for $a\not=1$, and $f_a=id$ iff $a=2$)
For instance, $f_4=\{1, 4, 3, 2, 5, 6, 7, 8, 9, 10, 11, 12,...\}$ and
$f_6=\{1, 6, 5, 2, 3, 4, 7, 8, 9, 10, 11, 12,...\}$. The sequence $f_4-f_6$ is $\{0, -2, -2, 0, 2, 2, 0, 0, 0, 0,...\}$ and continues to be zero.
Let's call two permutations $f_a$ and $f_b$, EI-permutations (eventually identical) if there exists $m$ which depends on $a$ and $b$ such that $f_a(n)=f_b(n)$ for all $n>m$. This (equivalence) relation partitions the set of bijections into equivalence classes, $\cal C$.
Q1: What is the set $J$ of all natural numbers such that for every $a\in J$ the permutations $f_a$ and $f_2=id$ are EI-permutations?
The answer seems to involve the primorial numbers https://en.wikipedia.org/wiki/Primorial
Q2: If $a,b\not\in J$, $f_a$ and $f_b$ appears to be EI-permutations. Show that $\cal C$ has only two classes: $[id]$ and $[f_3]$.
Q3: For $a\not\in J$, consider $\delta_n=|f_n(n+1)-f_a(n)|$. Show that this sequence is eventually a sequence containing only odd numbers.
This sequence appears to be connected with the Prime Gap sequence https://en.wikipedia.org/wiki/Prime_gap
Hi I am the proposer of the problem and nice to know that you have put some work into it.My solution(Proposer's solution) was based on the idea that the either set $f_a$ assumes the value of consequtive integers from some point onwards or there are numbers/elements in the set between which the elements are consequtive.I called such elements turning points and explored their properties