Related to this question, I wonder how often $n!+1$ is a prime?
There is a related OEIS sequence A002981, however, nothing is said if the sequence is finite or not. Does anybody know more about it?
$n! + 1$ is prime for $n = 0, 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, 26951, 110059, 150209, \dots$, no other factorial primes are known as of May 2014. See here for more info on factorial primes.
Just looking at the heuristics of the problem:
If you pick a random integer $x$, it will be a prime number with a probability about $1 / \ln x$. Now the number $n! + 1$ is not a random integer. We know that $n! + 1$ is not divisible by any prime number $p ≤ n$. A random large integer is not divisible by any prime $p ≤ n$ with probability $(1-1/2)(1-1/3)(1-1/5)...$ which is about $1 / (2 \ln n)$. So the likelihood that $n! + 1$ is a prime is accordingly higher, about $2 \ln n / \ln (n!)$.
Using the Stirling formula, $\ln (n!)$ is about $n \ln n - n$ or $n(\ln n - 1)$. So $n!+1$ is prime with probability about $(2/n)/(1 - 1 / \ln n)$.
The factor $(1 - 1 / \ln n)$ is quite close to 1; the number of primes of the form $n! + 1$ with $n ≤ M$ is about $2 \ln M$. Very roughly agrees with the list of primes given earlier (I think it is a list of known primes, with many numbers in between not examined).
Such numbers are called factorial primes. There is only limited number of known such numbers.
The largest factorial primes are discovered only recently. From an announcement of an organization called PrimeGrid PRPNet:
PrimeGrid is a set of projects based on distributed computing, and devoted to finding primes satisfying various conditions.
Factorial primes-related recent events in PrimeGrid:
$147855!-1$ found: official announcement
$110059!+1$ found: official announcement
$103040!-1$ found: official announcement
$94550!-1$ found: official announcement
Other current PrimeGrid activities: