A227537 Number of Mersenne primes that have between 10^n and 10^(n+1) - 1 digits.

7, 5, 6, 8, 5, 6, 7

Offset: 0

Author

Olivier de Mouzon, Jul 18 2013

Keywords

• nonn
• hard
• base

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.

Examples

			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.
		

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)

Crossrefs

Cf. A000043, A000668, A028335.