Whenever M_n = 2^n-1 is prime, the larger number P_n = (4^n - 2^n)/2 is "perfect", i.e. its factors add up to the number itself.

n=2: M_n = 3, P_n = 6 = 1 + 2 + 3

n=3: M_n = 7, P_n = 28 = 1 + 2 + 4 + 7 + 14

n=5: M_n = 31, P_n = 496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 +124 + 248

This new Mercenne prime makes for a new perfect number.

At a rough guess, the number of digits in a Mercenne prime is approx 30% of n, and the number of digits in the corresponding perfect number is twice this, approx 60% of n.

(Exercise for the reader: prove this.)

*Edited: 16 Feb 2013, 11:08 a.m. *