[vsogakpo] n^n and Mersenne numbers

Mersenne numbers grow as approximately n^n, where n is the index into the sequence.  One doesn't often see n^n "in real life" because dimensions (units) become weird.  (Sometimes we see it in combinatorics.)

M(n) = 2^nthprime(n)-1 ~= 2^(n*log(n)) by Prime Number Theorem
= exp(n*log(n)*log(2))
= n^(log(2)*n) = (n^log(2))^n = (n^n)^log(2)

Incidentally, by applying PNT again to the second line, the probability the nth Mersenne number is prime is 1/(n*log(n)*log(2)).  The integral of that expression diverges, suggesting there are an infinite number of Mersenne primes.