A028335 Number of decimal digits in n-th Mersenne prime.

1, 1, 2, 3, 4, 6, 6, 10, 19, 27, 33, 39, 157, 183, 386, 664, 687, 969, 1281, 1332, 2917, 2993, 3376, 6002, 6533, 6987, 13395, 25962, 33265, 39751, 65050, 227832, 258716, 378632, 420921, 895932, 909526, 2098960, 4053946, 6320430, 7235733, 7816230, 9152052, 9808358, 11185272

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• base

Examples

			A000668(6) = 2^17-1 = 131071 has 6 decimal digits, so a(6) = 6.
A000668(10) = 2^89-1 = 618,970,019,642,690,137,449,562,111 has 27 digits, so a(10) = 27.
		

References

• Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 19.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..48 (terms 1..47 from Ivan Panchenko)
• Chris K. Caldwell, Mersenne Primes.
• GIMPS, List of Known Mersenne Prime Numbers.
• Romeo Meštrović, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2023.

Crossrefs

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

Cf. A000668, A055642.

Programs

Maple

seq(length(numtheory:-mersenne([i])),i=1..45); # Robert Israel, Feb 02 2018

Mathematica

IntegerLength[2^Array[MersennePrimeExponent, 45] - 1] (* Jean-François Alcover, Feb 17 2018 *)
a[n_] := Floor[MersennePrimeExponent[n]/Log2[10]] + 1; Array[a, 48] (* Amiram Eldar, Oct 16 2024 *)

Formula

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

a(n) = A055642(A000668(n)). - Michel Marcus, Apr 07 2018

Extensions

More terms from Enoch Haga, Dec 18 2001

a(38) from Harry J. Smith, Apr 17 2003

a(39) from Omar E. Pol, Oct 28 2007

a(40)-a(41) from Jason Kimberley, Jan 05 2012

a(42)-a(45) from Patrick J. McNab, Feb 01 2018