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%I #22 Nov 02 2013 22:02:44
%S 7,5,6,8,5,6,7
%N Number of Mersenne primes that have between 10^n and 10^(n+1) - 1 digits.
%C 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].
%C 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.
%C 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.
%C 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.
%H GIMPS, <a href="http://www.mersenne.org/">Great Internet Mersenne Prime Search official Home Page</a> and <a href="http://www.mersenne.org/report_milestones/default.php">Great Internet Mersenne Prime Search milestones</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Great_Internet_Mersenne_Prime_Search">Great Internet Mersenne Prime Search</a> or more up to date the French version: <a href="http://fr.wikipedia.org/wiki/Great_Internet_Mersenne_Prime_Search">Great Internet Mersenne Prime Search (FR)</a>
%e For n = 1, a(n) = 5 Mersenne primes with 10 to 99 digits, which are:
%e * M8 = M_31 = 2147483647,
%e * M9 = M_61 = 2305843009213693951,
%e * M10 = M_89 = 618970019642690137449562111,
%e * M11 = M_107 = 162259276829213363391578010288127,
%e * M12 = M_127 = 170141183460469231731687303715884105727.
%Y Cf. A000043, A000668, A028335.
%K nonn,hard,base
%O 0,1
%A _Olivier de Mouzon_, Jul 18 2013
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