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 bits in binary expansion of n-th Mersenne prime (A117293). - Artur Jasinski (grafix(AT)csl.pl), Feb 09 2007

Number of divisors of n-th even perfect number, divided by 2. Number of divisors of n-th even perfect number that are powers of 2. Number of divisors of n-th even 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), assuming there are no odd superperfect numbers. - Omar E. Pol (info(AT)polprimos.com), Mar 01 2008

Differences between exponents when the even perfect numbers are represented as differences of powers of 2, for example: The 5th even perfect number is 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

Number of 1's in binary expansion of n-th even perfect number (See A135650). Number of 1's in binary expansion of divisors of n-th even perfect number that are multiples of n-th Mersenne prime A000668(n) (See A135652, A135653, A135654, A135655). - Omar E. Pol (info(AT)polprimos.com), May 04 2008

Indices of the numbers A006516 that are also even perfect numbers. [From Omar E. Pol (info(AT)polprimos.com), Aug 30 2008]

Indices of Mersenne numbers A000225 that are also Mersenne primes A000668. [From Omar E. Pol (info(AT)polprimos.com), Aug 31 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.

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

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

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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.

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

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

H. Lifchitz, Mersenne and Fermat primes field

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

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

K. Schneider, PlanetMath.org, Mersenne numbers

H. J. Smith, Mersenne Primes

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, MathWorld Headline News, 47th Known Mersenne Prime Apparently Discovered [From Lekraj Beedassy (blekraj(AT)yahoo.com), Aug 03 2009]

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, 42ndMersenne 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

Eric Weisstein, MathWorld Headline News, 45th and 46th Mersenne Primes Found [From Lekraj Beedassy (blekraj(AT)yahoo.com), Sep 18 2008]

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

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

A000043(n)=Log[(1/2)(1+Sqrt[1+8*A000396(n)])]/Log[2] [From Artur Jasinski (grafix(AT)csl.pl), Sep 23 2008]

a(n) = A000005(A061652(n)). [From Omar E. Pol (info(AT)polprimos.com), Aug 26 2009]

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 ...

Select[Range[10^3], PrimeQ[2^#-1]&] - Vladimir Orlovsky (4vladimir(AT)gmail.com), Apr 29 2008

(PARI) isA000043(n) = isprime(2^n-1) [From Michael Porter (michael_b_porter(AT)yahoo.com), Oct 28 2009]

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, A000225, A000396, A090748, A133033, A135655, A006516, A019279, A061652, A075398, A133033, A135650, A135652, A135653, A135654.

Cf. A000005. [From Omar E. Pol (info(AT)polprimos.com), Aug 26 2009]

Sequence in context: A136003 A123856 A120857 this_sequence A109799 A152961 A109461

Adjacent sequences: A000040 A000041 A000042 this_sequence A000044 A000045 A000046

hard,nonn,nice,core

N. J. A. Sloane (njas(AT)research.att.com).

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

Also in the sequence: 2^37156667 - 1 and 2^43112609 - 1. - Sep 15 2008

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). - N. J. A. Sloane (njas(AT)research.att.com), Dec 30 2005