Let integers $\ a>1\ $ and $\ b\in\mathbb Z\ $ be relatively prime (hence $\ b\ne 0).\ $ The Dirichlet's prime distribution theorems apply to the arithmetic sequence

$$ (_aG_b(x) : x\in\mathbb Z) $$ where $$ _aG_b(x)\ :=\ a\cdot x + b\ $$

Thus, for arbitrary integer $\ s>0\ $ that is relatively prime to $\ b,\ $ I'd like to see some prime distribution theorems for the nearly geometric sequence

$$ \left(\left(\bigcirc^n\,_aG_b\right)(s): \ s\in\mathbb Z_{_{\ge0}} \right) $$

where $\ \bigcirc^n\ $ is my notation for the composition power (*please, let be, do not "correct it"*). Can you prove any? It must be difficult but one may try to prove **comparison** prime distribution theorems where one **compares** the distribution of primes of two nearly geometric sequences (even **without** knowing anything specific about any single sequence like this).

Let

$$\ \gamma\ :=\ \frac b{a-1} $$

Then $$ _aG_b(x)\ =\ a\cdot(x+\gamma)-\gamma $$ and $$ \forall_{n\in\mathbb Z_{_{\ge0}}}\quad \left(\bigcirc^n\,_aG_b\right)(s)\ =\ a^n\cdot(x+\gamma)-\gamma $$

Thus, we may rewrite our nearly geometric series as

$$ \left(a^n\!\cdot\!(x+\gamma)\,-\,\gamma\,:
\ \ n\in\mathbb Z_{_{\ge0}}\right) $$
(making it more *geometric*).

**Examples**:

$\ \left(\bigcirc^n\,_2G_1\right)(1)\ $ -- the $n$-th Mersenne number;

$\ \left(\left(\bigcirc^n\,_2G_1\right)(p): \ p\in\mathbb Z_{_{\ge0}} \right)\ $ -- a potential source for Sophie Germain primes (in particular when $\ p\ :=\ _2G_1(s)\ $ is prime, while natural number $\ s\ $ is not prime)

Of course, one would like to know as much as possible about subsequences of consecutive prime terms of nearly geometric sequences.