distribution of prime numbers in special range

Question

Let's start from the first prime $2$,

primes from $2+1$ to $2^2+1$ is $3, 5$,

primes from $5+1$ to $5^2+1$ is $7, 11, 13, 17, 19, 23$ ($6$ primes)

primes from $23+1$ to $23^2+1$ is $29, ... 523$ ($90$ primes)

primes from $523+1$ to $523^2+1$ is $541, ... 273527$ ($23826$ primes)

here we get the sequence $1, 2, 6, 90, 23826$, the question is from $23826$ on ,are there all even numbers?

Answer 1

There are not all even numbers, the sequence is $2, 6, 90, 23826, 3118310544, 114107585701006770667, a_6$ calculated by Kim with his primecount project on Amazon Cloud.