**Definition - Denizen**

A sequence $\lbrace a_k \rbrace$ is a

denizenif all of it's members are prime numbers, i.e $a_0, a_1, ... a_n \in \mathbb{P} $; and it satisfies the following condition;if"$a_{x_1} =y_1 $", "$a_{x_2} =y_2$", "$x_1 \pm m_1y_1 \neq x_2 \pm m_2 y_2 $ when $y_2<y_1 $" and "$m_3$ isn't divisible by $y_1$";then"$a_{x_1 \pm m_1y_1}=y_1$" and $"a_{x_1 \pm m_3} \neq y_1$" (where $m \in\mathbb{N}$ where $y \in\mathbb{P}$ and where $x \in\mathbb{Z}$).

Let a denizen consisting of prime numbers up to and including $p_\alpha$ be denoted $\lbrace a_k \rbrace ^{p_\alpha} $. For example; a denizens that can be denoted as $\lbrace a_k \rbrace^7 $ is {2,7,2,3,2,5,2}.

**Question**

What is the maximum length $\lbrace a_k \rbrace ^{p_\alpha} $ can take?

**Attempt**

In order to find the maximum length a denizen $\lbrace a_k \rbrace ^{p_\alpha} $ can take I considered denizens of two different types.

I first considered a denizen $\lbrace a_k \rbrace ^{p_\alpha} $ of a type which corresponds to the sequence of natural numbers $ \lbrace 2,3,4,...,p_{\alpha +1} -1 \rbrace $. The corresponding denizen is $\lbrace 2,3,2,...,2 \rbrace $. This type of denizen had been created such that $a_i=d_i|i$, where $d_i$ is some divisor of $i$, implies $a_i \in $$\lbrace a_k \rbrace$. The consequence of this property is that this type of denizen can be considered a sequence of the lowest prime divisors of the natural numbers from $2$ to $p_{\alpha +1} -1$ respectively. For example, of this type; $\lbrace a_k \rbrace ^{11} = \lbrace 2,3,2,5,2,7,2,3,2,11,2 \rbrace $ and corresponds to $\lbrace 2,3,4,5,6,7,8,9,10,11,12 \rbrace$.

A denizen $\lbrace a_k \rbrace ^{p_\alpha} $ of this type which has a length of $ p_{\alpha +1} -2 $.

However I found a larger type of denizen corresponding to the sequence of integers $\lbrace -(p_{\alpha -1} -1), ..., -4,-3,-2,-1,0,1,2,3,4, ..., p_{\alpha -1} -1 \rbrace $. The corresponding denizen is $\lbrace 2, ..., 2,3,2,p_\alpha,2,p_{\alpha -1},2,3,2, ..., 2 \rbrace $. This type of denizen can also be considered a sequence of lowest prime factors of the integer sequence above, however it also requires the replacement of $-1$ and $1$ with the two primes $p_{\alpha}$ and $p_{\alpha -1}$ respectively and involves replacing $0$ with $2$. For example, of this type $\lbrace a_k \rbrace ^{11} = \lbrace 2,5,2,3,2,7,2,11,2,3,2,5,2 \rbrace $ and corresponds to $\lbrace -6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6 \rbrace$.

This type of denizen has a length of $ 2p_{\alpha -1} -1 $ and also has an apparent symmetry as defined below. this type of denizen has a length greater or equal to the length of the previous type because of the identity $2p_{x-2} \geq p_{x}-1$ proved here.

So my next question is; is the second type of denizen the largest lengthed $\lbrace a_k \rbrace ^{p_\alpha} $ denizen possible? How could you prove it was?

I have tried to prove this, using the concept of symmetry.
I defined the **symmetric depth** as the largest prime number $p_N$ such that $a_{x+ p_N}=a_{x- p_N} = p_N$ and $a_{x+ p_{i}}=a_{x- p_{i}} =p_i$ for all prime numbers $p_i$ less than $p_N$, centred on some $a_x\in \lbrace a_k \rbrace$. I let $\lbrace a_k \rbrace ^{p_\alpha} _ {p_N} $ be a denizen consisting prime numbers upto and including $p_\alpha$ with symmetric depth $p_N$, and hoped to prove that the length of a denizen $\lbrace a_k \rbrace ^{p_\alpha} _ {p_N} $ is always greater than or equal to the length of a denizen $\lbrace a_k \rbrace ^{p_\alpha} _ {p_{N-1}} $. However I found a counter example to this, due to the fact that the gaps between large prime numbers tend to be greater than the gaps between smaller prime numbers.

However I have yet to produce a denizen $\lbrace a_k \rbrace ^{p_\alpha} $ greater in length than $\lbrace a_k \rbrace ^{p_\alpha} _ {p_{\alpha -2}} $.

So any ideas to further this?