Any series $\displaystyle \sum_{k=0}^{\infty}a_k2^{-k}$, where $a_k\in\{0,1\}$, converges to some $x\in[0,2]$ and since the sequence $a_n$ is unique for each $x\in[0,2]$ there is an bijection between $\text{func}(\mathbb{N},\{0,1\})$ and $[0,2]$.
Is there any known sequence $a_n\in\text{func}(\mathbb{N},\{0,1\})$ (or real number $x\in[0,2]$) such that for any $n\in \mathbb{N}$ there is an $m>n$ such that the binary pattern $a_0a_1\cdots a_m$ corresponds to a prime? That is, does it exists such a sequence so that $\displaystyle \sum_{k=0}^{m}a_k2^{m-k}$ is a prime for infinitely many m?
The Mersenne primes correspond to $x=2$.
It seems like orangeskid gave me the answer in comments but I didn't understand it first (or second, or third):
From results about disjunctive sequences it is clear that any binary string $a_0a_1\cdots a_m$ corresponding to a prime can be extended to a bigger string corresponding to a prime. Therefore it exists $x\in[0,2]$ that is related to the primes in this way.
And from Dirichlet's theorem it follows that any $x\in[0,2]$ can be arbitrarily well approximated by a sequence $a_0a_1\cdots a_m$ and then be completed to a prime string as above, repeatedly.
$\therefore$ The set of all prime related numbers in $[0,2]$ is dense.
So I reverse the question, are there non trivial examples of
$\displaystyle \sum_{k=0}^{\infty}a_k2^{-k}\in[0,2]$ for which there is a maximal prime $\displaystyle \sum_{k=0}^{m}a_k2^{m-k}$?
