I searched for primes of the form of $\lfloor \zeta(-n) \rfloor$, where $n \in \Bbb{N}$, for a range of $n \le 10^4$ on PARI/GP and found $\lfloor \zeta(-n) \rfloor$ is only prime for $n=23$.
My PARI code:
for(n=1, 10^4, if(ispseudoprime(floor(-bernfrac(2*n)/(2*n)))==1, print([2*n-1, floor(-bernfrac(2*n)/(2*n))])); print(2*n-1))
Note that $\zeta(-n)$ for odd $n$ (for even $n$, $\zeta(-n)=0$) can be also expressed as:
$$\zeta(-(2n-1))=-\frac{B_{2n}}{2n}$$
where $B_{2n}$ is the $2n$th Bernoulli number and written as brenfrac(2*n) in PARI/GP.
Questions:
$(1)$ Is $\lfloor \zeta(-n) \rfloor$, where $n \in \Bbb{N}$, only prime for $n=23$?
$(2)$ Are there finite primes of the form of $\lfloor \zeta(-n) \rfloor$?
I would appreciate any counterexamples(can be a probable prime)/proofs/papers.
Extra: I also searched for primes of the form $\lceil\zeta(-n)\rceil$, $\lfloor\zeta(-n)\rceil$, $\lceil B_n \rceil$ and $\lfloor B_n \rceil$. For $n \le 10^4$:
$(1)$ $\lceil\zeta(-n)\rceil$ seems to be only prime for $n=691$.
$(2)$ There seems to be no prime of the form $\lfloor\zeta(-n)\rceil$.
$(3)$ $\lceil B_n \rceil$ seems to be only prime for $n=14$.
$(4)$ $\lfloor B_n \rceil$ seems to be only prime for $n=38$.