I made a conjecture, but donâ€™t know how to prove it. Perhaps it could be related to Mersenne Primes?

Consider

.

If

then

.

This is to say, if a natural number (positive integer)

is equal to the product of the 1st prime, the 2nd prime, the 3rd prime, and so on, until the k-th prime, then the amount of all the divisors of

(including 1 and

) will be equal to

.

How must we go about proving this? If we do, perhaps we could build an algorithm to find a prime value for k such that

is prime.