The proof can be further simplified by noticing that the primorial (a term coined by Harvey Dubner) in the denominator is an even integer while the numerator is an odd integer.
By the way, one could possibly use the term 'Mersennerial', n#_{M}, for the product of the first n Mersenne prime exponents.
For instance, 13#_{M} = 2 × 3 × 5 × 7 × 13.
Also, π_{M}(n) could be the Mersenneprimeexponentcounting function, and π_{M}(13) = 5.
