TITLE: Generalization of Bertrand's Postulate
QUESTION [11 upvotes]: Bertrand's postulate states that there is a prime $p$ between $n$ and $2n-2$ for $n>3$. According to Dirichlet's theorem we have that a sequaence
$$a\cdot n+b$$
has infinite primes iff $a$ and $b$ are relatively prime. So in some sense, Bertrand's postulate gives a maximum of time for encountering a prime in the sequence
$$2\cdot n+1$$
So, the question is: there is a generalization of Bertrand's Postulate for sequences $a\cdot n+b$ that accomplish the Dirichlet's theorem?
EDIT: (For a more concise explanation of the particular generalization.) We know that given 
$$a_n=2\cdot n+1$$
we have that for all $m$ there is a prime in the sequence greater than $a_m$ and less than $a_{2m}$. So, the thing is that if there is some generalization of Bertrand's Postulate using the sequence form, for an arbitrary sequence
$$c_n=a\cdot n+b$$
with $a$ and $b$ coprime. Something as, for every relatively prime $a$ and $b$, there is a $k\leq a\cdot b$, such that for all $m$ there is a prime in the sequence between $c_m$ and $c_{k\cdot m}$.
Such kind of thing is what I am looking for.

REPLY [2 votes]: Here is another generalization
Let $n$ and $k$ be positive integers such that $n > 1.1\log(2.5k)$ then there are at least $k-1$ primes between $n$ and $kn$.
Ref: https://arxiv.org/pdf/1710.09891.pdf