# The longest sequence of numbers with a certain divisibility property

EDIT- result. westzynthius(1931) showed that we can create a $p_x$ denizen longer than $p_x \times log(log(log(p_x)))$... Meaning for large enough prime numbers, the maximum denizen is much larger than $2p_x, 3p_x$ etc.

Definition - Denizen

A sequence $\lbrace a_k \rbrace$ is a denizen if 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.