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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)*lnlog(2) + lnlog(lnlog(10)/lnlog(2))*lnlog(2)/lnlog(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 [lnlog(lnlog(x))]^2/lnlog(x). Hence primes are still around 6*[lnlog(lnlog(x))]^2 more frequent among Mersenne numbers than among numbers.

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nonn,hard,changed,base

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At the moment (Jul 18 July 2013), there are already 4 Mersenne primes in the next group (n = 7), the last one was discovered on Jan 25 January 2013 and has 17,425,170 17425170 digits.

Note that for n = 6, a(n) = 7 still needs full confirmation, as tests for all factors between M42 = M_25,964,951 25964951 and M_44,457,869 44457869 (more than 10^7 digits) have only made once and a double check is needed to confirm a(6) = 7.

Number of Mersenne primes that have between 10^n and 10^(n+1) - 1 digits, starting n = 0.

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>

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>

* M8 = M_31 = 2147483647,

* M9 = M_61 = 2305843009213693951,

* M10 = M_89 = 618970019642690137449562111,

* M11 = M_107 = 162259276829213363391578010288127,

* M12 = M_127 = 170141183460469231731687303715884105727.

Cf. A000043, A000668, A028335,A000043.

nonn,nice,hard,changed

proposed

editing

editing

proposed

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)*ln(2) + ln(ln(10)/ln(2))*ln(2)/ln(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 [ln(ln(x))]^2/ln(x). Hence primes are still around 6*[ln(ln(x))]^2 more frequent among Mersenne numbers than among numbers.

proposed

editing