1,1

It is believed (but unproved) that this sequence is infinite. The data suggests that the number of terms up to exponent N is roughly K log N for some constant K.

Length of prime repunits in base 2.

The associated perfect number N=2^(p-1)*M(p) (=A019279*A000668=A000396), has 2p (=A061645) divisors with harmonic mean p (and geometric mean sqrt(N)). - Lekraj Beedassy (blekraj(AT)yahoo.com), Aug 21 2004

In one of his first publications Euler found the numbers up to 31 but erroneously included 41 and 47.

Equals number of digits in binary expansion of n-th Marsenne prime (A117293). - Artur Jasinski (grafix(AT)csl.pl), Feb 09 2007

Number of divisors of n-th perfect number A000396(n), divided by 2. Number of divisors of n-th perfect number that are powers of 2. Number of divisors of n-th perfect number that are multiples of n-th Mersenne prime A000668(n). - Omar E. Pol (info(AT)polprimos.com), Feb 24 2008

Number of divisors of n-th even superperfect number A061652(n). Numbers of divisors of n-th superperfect number A019279(n), if there are no odd superperfect numbers. - Omar E. Pol (info(AT)polprimos.com), Mar 01 2008

Differences between exponents when the perfect numbers are represented as differences of powers of 2, for example: A000396(5)=33550336=2^25 - 2^12 then a(5)=25-12=13 (See A135655, A133033, A090748). - Omar E. Pol (info(AT)polprimos.com), Mar 01 2008

Base 2 logarithm of (1 + n-th Mersenne prime A000668(n)). - Omar E. Pol (info(AT)polprimos.com), Mar 02 2008

Base 2 logarithm of A075398(n). - Omar E. Pol (info(AT)polprimos.com), Apr 17 2008

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 4.

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.

G. Everest et al., Primes generated by recurrence sequences, Amer. Math. Monthly, 114 (No. 5, 2007), 417-431.

F. Lemmermeyer, Reciprocity Laws From Euler to Eisenstein, Springer-Verlag, 2000, p. 57.

B. Tuckerman, The 24th Mersenne prime, Notices Amer. Math. Soc., 18 (Jun, 1971), Abstract 684-A15, p. 608.

B. Tuckerman, The 24th Mersenne prime, Proc. Nat. Acad. Sci. USA, 68 (1971), 2319-2320.

K. Zsigmondy, Zur Theorie der Potenreste, Monatsh. Math., 3 (1892), 265-284.

Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 19.

David Wasserman, Table of n, a(n) for n = 1..39

J. Bernheiden, Mersenne Numbers (Text in German)

Andrew R. Booker, The Nth Prime Page

J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.

P. G. Brown, A Note on Ramanujan's (FALSE) Conjectures Regarding 'Mersenne Primes'

C. K. Caldwell, Mersenne Primes

C. K. Caldwell, Recent Mersenne primes

L. Euler, Observations on a theorem of Fermat and others on looking at prime numbers

L. Euler, Observationes do theoremate quodam Fermatiano aliisque ad numeros primos spectantibus

GIMPS (Great Internet Mersenne Prime Search), Distributed Computing Projects

GIMPS (Great Internet Mersenne Prime Search), Current status of search

Wilfrid Keller, List of primes k.2^n - 1 for k < 300

A. J. Menezes, P. C. van Oorschot and S. A. Vanstone, Handbook of Applied Cryptography, CRC Press, 1996; see p. 143.

G. P. Michon, Perfect Numbers, Mersenne Primes

M. Oakes, A new series of Mersenne-like Gaussian primes

K. Schneider, PlanetMath.org, Mersenne numbers

H. S. Uhler, On All Of Mersenne's Numbers Particularly M_193

H. S. Uhler, First Proof That The Mersenne Number M_157 Is Composite

S. S. Wagstaff, Jr., The Cunningham Project

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Mathworld Headline News, 40-th Mersenne Prime Announced

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Mathworld Headline News, 41st Mersenne Prime Announced

Eric Weisstein, MathWorld Headline News, 42nd Mersenne Prime Found

Eric Weisstein's World of Mathematics, Integer Sequence Primes

Eric Weisstein, MathWorld Headline News, 43rd Mersenne Prime Found

Eric Weisstein, MathWorld Headline News, 44th Mersenne Prime Found

David Whitehouse, Number takes prime position (2^13466917 - 1 found after 13000 years of computer time)

George Woltman et al., GIMPS - The Great Internet Mersenne Prime Search

Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime

O. E. Pol, Determinacion geometrica de los numeros primos y perfectos.

H. J. Smith, Mersenne Primes

H. Lifchitz, Mersenne and Fermat primes field

Corresponding to the initial terms 2, 3, 5, 7, 13, 17, 19, 31 ... we get the Mersenne primes 2^2 - 1 = 3, 2^3 - 1 = 7, 2^5 - 1 = 31, 127, 8191, 131071, 524287, 2147483647 ...

See A000668 for the actual primes.

Cf. A001348, A016027, A046051, A057429, A057951-A057958, A066408.

Cf. also A117293, A127962, A127963, A127964, A127965, A127961, A000979, A000978, A124400, A124401, A127955, A127956, A127957, A127958, A127936.

Cf. also A134458.

Cf. A133033.

Cf. A019279, A061652.

Cf. A000396, A090748, A133033, A135655.

Adjacent sequences: A000040 A000041 A000042 this_sequence A000044 A000045 A000046

Sequence in context: A136003 A123856 A120857 this_sequence A109799 A109461 A090422

Cf. A075398.

hard,nonn,nice,core

njas

2^6972593 - 1 is known to be the 38th Mersenne prime. - Harry J. Smith (hjsmithh(AT)sbcglobal.net), Apr 17 2003

2^13466917 - 1 is known to be the 39th Mersenne prime.

Also in the sequence: 2^20996011 - 1 (a 6.3 million digit number). - Nov 17, 2003. See the GIMPS link for details.

Also in the sequence: 2^24036583 - 1 (a 7.2 million digit number). - Jun 01, 2004

Also in the sequence: 2^25964951 - 1 (a 7.8 million digit number). - Feb 26, 2005

Also in the sequence: 2^30402457 - 1 (a 9.2 million digit number). - Dec 29, 2005

Also in the sequence: 2^32582657 - 1. - Sep 21 2006

As of Dec 30 2005 the exhaustive search been run through 16693000, according to the GIMPS status page (thanks to R. K. Guy for this information). - njas, Dec 30 2005