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.
Best Answer
One generalization of Bertrand's postulate I know is a theorem of Sylvester and Schur. See for example http://www.math.uiuc.edu/~pppollac/sschur.pdf
The theorem says that for any positive integer $k$ the product of $k$ consecutive integers greater than $k$ contains a prime factor greater than $k$.
I hope this helps you somehow.