Is there an easier prime factorization method for the sum of a prime's powers?

I need to obtain prime factorizations of numbers of the type: $sum_{i=0}^n p^i$, for any prime number $p$ (not the same one each time).

Do you know if there is a quicker algorithm to calculate these factorizations than those used for other natural numbers?

I don’t know if there is a known solution. My only lead is that all Mersenne primes are of the form $sum_{i=0}^n 2^i$.

Edit: by prime factorization I mean, for example, if $p$ is 3 and $n$ is 6, the number is 364, and the prime factorization I’m looking for is 2^2, 7 and 13.