%I N0189 #68 Oct 16 2024 09:22:29

%S 1,1,2,3,4,6,6,10,19,27,33,39,157,183,386,664,687,969,1281,1332,2917,

%T 2993,3376,6002,6533,6987,13395,25962,33265,39751,65050,227832,258716,

%U 378632,420921,895932,909526,2098960,4053946,6320430,7235733,7816230,9152052,9808358,11185272

%N Number of decimal digits in n-th Mersenne prime.

%D Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 19.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%H Amiram Eldar, <a href="/A028335/b028335.txt">Table of n, a(n) for n = 1..48</a> (terms 1..47 from Ivan Panchenko)

%H Chris K. Caldwell, <a href="http://www.utm.edu/research/primes/mersenne/index.html">Mersenne Primes</a>.

%H GIMPS, <a href="https://www.mersenne.org/primes/">List of Known Mersenne Prime Numbers</a>.

%H Romeo Meštrović, <a href="http://arxiv.org/abs/1202.3670">Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof</a>, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2023.

%F a(n) = floor(A000043(n)*log(2)/log(10)) + 1.

%F a(n) = A055642(A000668(n)). - _Michel Marcus_, Apr 07 2018

%e A000668(6) = 2^17-1 = 131071 has 6 decimal digits, so a(6) = 6.

%e A000668(10) = 2^89-1 = 618,970,019,642,690,137,449,562,111 has 27 digits, so a(10) = 27.

%p seq(length(numtheory:-mersenne([i])),i=1..45); # _Robert Israel_, Feb 02 2018

%t IntegerLength[2^Array[MersennePrimeExponent, 45] - 1] (* _Jean-François Alcover_, Feb 17 2018 *)

%t a[n_] := Floor[MersennePrimeExponent[n]/Log2[10]] + 1; Array[a, 48] (* _Amiram Eldar_, Oct 16 2024 *)

%Y See A000043, which is the main entry for this sequence.

%Y Cf. A000668, A055642.

%K nonn,base

%O 1,3

%A _N. J. A. Sloane_

%E More terms from _Enoch Haga_, Dec 18 2001

%E a(38) from _Harry J. Smith_, Apr 17 2003

%E a(39) from _Omar E. Pol_, Oct 28 2007

%E a(40)-a(41) from _Jason Kimberley_, Jan 05 2012

%E a(42)-a(45) from _Patrick J. McNab_, Feb 01 2018