Can this sequence be finite for prime numbers?!

A certain Petrov I. B. (Ivan Borisovich, he has a namesake - Igor Borisovich) published an article "[Petrov I. B. "Quasi-exponential primes", SI, 2021] (only in Russian, for now), where he proposed to consider a sequence of numbers of the form a^a-a-1, where a > 2, a is any natural number, to search for large primes. The publication says that he calculated the numbers up to a = 5000. The largest a at which the Petrov's number is prime a = 1379.

I was interested in the topic purely from the practical side of the question - the search for new large primes. I have not yet figured out why Petrov offers such a strange formula, but I assume that this is not just so. Subconsciously , there is something in it, some meaning.

I used a small program that checks the numbers using the Miller-Rabin algorithm for 9 rounds. And I ran the "Petrov's numbers" through it to a = 10,000 - zero result. If it's not a mistake. Why is there such a spread of variable values? The last found value is a = 1379. The following is clearly a > 10,000.

In general, I'm going to use this formula in my program, where prime numbers are used. Why am I still interested: a series of values of the variable a for which the Petrov's numbers are simple a = 3, 4, 5, 6, 9, 17, 22, 85, 710, 844, 1379. (Judging by the author's publication, I did not check it myself). And then there is nothing at least until a = 10,000. A very strange probability distribution of prime numbers for almost exponentially increasing values of numbers...

Can this sequence be finite for prime numbers?!