I was playing around with the sequence where the $k^{th}$ number is equal to $2^k-1$. It seems that all numbers except $63$ contain at least one new prime in there prime factorization. That is a prime that has not occurred in the prime factorization of any of the previous numbers in the sequence. Is there a reason why this is or can anyone find a counter example? I've been trying in mathemtica but my programming ability is sub-par.
This sequence is also generated by starting with $1$ and each term is $2k+1$ the previous term.
Note I do not care whether the term itself is prime (Mersenne primes) but that each new term contains a new prime factor.
You can use Zsigmondy's theorem (see Zsigmondy's theorem), which proves what you observed : except for 63, every term of your sequence has a prime factor that does not divide the previous terms.