OFFSET
0,1
COMMENTS
The nice property of this sequence is that (at least up to n = 6) there seems to be a rather stable number of Mersenne primes for each digit number group [10^n ... 10^(n+1) - 1].
At the moment (Jul 18 2013), there are already 4 Mersenne primes in the next group (n = 7), the last one was discovered on Jan 25 2013 and has 17425170 digits.
Note that for n = 6, a(n) = 7 still needs full confirmation, as tests for all factors between M42 = M_25964951 and M_44457869 (more than 10^7 digits) have only made once and a double check is needed to confirm a(6) = 7.
If this sequence were to actually be stable, this would mean that the number of Mersenne primes having between 10^n and 10^(n+1) - 1 digits is always around 6, when the number of prime numbers in the same digit number group constantly increases: around 2.3*10^(10^(n+1)-(n+1)). Also the number of Mersenne numbers in the same digit group constantly increases (though much less than the number of prime numbers): 9*10^n/[(n+1)*log(2) + log(log(10)/log(2))*log(2)/log(10)]. So, if a(n) is really rather stable (around 6), Mersenne primes frequency among Mersenne numbers lower than x is converging towards 0 in the magnitude of [log(log(x))]^2/log(x). Hence primes are still around 6*[log(log(x))]^2 more frequent among Mersenne numbers than among numbers.
LINKS
GIMPS, Great Internet Mersenne Prime Search official Home Page and Great Internet Mersenne Prime Search milestones
Wikipedia, Great Internet Mersenne Prime Search or more up to date the French version: Great Internet Mersenne Prime Search (FR)
EXAMPLE
For n = 1, a(n) = 5 Mersenne primes with 10 to 99 digits, which are:
* M8 = M_31 = 2147483647,
* M9 = M_61 = 2305843009213693951,
* M10 = M_89 = 618970019642690137449562111,
* M11 = M_107 = 162259276829213363391578010288127,
* M12 = M_127 = 170141183460469231731687303715884105727.
CROSSREFS
KEYWORD
nonn,hard,base
AUTHOR
Olivier de Mouzon, Jul 18 2013
STATUS
approved