TITLE: Examples of other Mills' constants
QUESTION [5 upvotes]: Mills' constant is the well-known constant A such that the function $\lfloor A^{{3}^{n}} \rfloor$ gives primes for all natural numbers $n$, whose value is ~$1.306377883...$ 
It is also well-known that there is an infinity of functions $\lfloor A^{{r}^{n}} \rfloor$ that gives primes for all $n$, depending on the value of $r$.
Are there any examples of such $\lfloor A^{{r}^{n}} \rfloor$ where $r \neq 3$? There seems to be very little information available on this subject.

REPLY [3 votes]: Given a particular value of $r \ge 3$, a high precision calculator and a method of finding the next prime above a number, it is easy to find the initial digits of $A_r$. 
For example with $r=4$: 

start with the smallest prime $2$  
$2^4=16$
the smallest prime above $16$ is $17$
$17^4 = 83521$ 
the smallest prime above $83521$ is $83537$
etc.

We now want $2 \le A_4^{4^1} \lt 3$ and $17 \le A_4^{4^2} \lt 18$ and $83537 \le A_4^{4^3} \lt 83538$ and so on
This will give $A_4=1.193725\ldots$
Meanwhile, for $r=2$, then if Legendre's conjecture that there is a prime between $n^2$ and $(n+1)^2$ is true,  you have $\lfloor 1.5246999605380943599233635756884211622202236231…^{2^n}\rfloor$ giving primes: see OEIS A059784