1,1

Equivalently, primes of form 2^n - 1 for integers n.

See A000043 for the values of p.

Prime repunits in base 2.

Define f(k) = 2k+1; begin with k = 2, a(n+1) = least prime of the form f(f(f(...(a(n)))). - Amarnath Murthy, Dec 26 2003

Mersenne primes other than the first are of form 6n+1. - Lekraj Beedassy, Aug 27 2004

A034876(a(n)) = 0 and A034876(a(n)+1) = 1. - Jonathan Sondow, Dec 19 2004

Mersenne primes are solutions to sigma(n+1)-sigma(n)=n as perfect numbers (A000396(n)) are solutions to sigma(n)=2n. - Benoit Cloitre, Feb 07 2002

Appears to give all n such that sigma(n+1)-sigma(n)=n. - Benoit Cloitre, Aug 27 2002

If n is in the sequence then sigma(sigma(n))=2n+1. Is it true that this sequence gives all numbers n such that sigma(sigma(n))=2n+1? - Farideh Firoozbakht, Aug 19 2005

Mersenne primes other than the first are of form 24n+7; see also A124477. - Artur Jasinski, Nov 25 2007

It is easily proved that if n is a Mersenne prime then sigma(sigma(n)) - sigma(n) = n. Is it true that Mersenne primes are all the solutions of the equation sigma(sigma(x)) - sigma(x) = x? - Farideh Firoozbakht, Feb 12 2008

Sum of divisors of n-th even superperfect number A061652(n). Sum of divisors of n-th superperfect number A019279(n), if there are no odd superperfect numbers. - Omar E. Pol, Mar 11 2008

Indices of both triangular numbers and generalized hexagonal numbers (A000217) that are also even perfect numbers. - Omar E. Pol, May 10 2008, Sep 22 2013

Number of positive integers (1, 2, 3,...) whose sum is the n-th perfect number A000396(n). - Omar E. Pol, May 10 2008

Vertex number where the n-th perfect number A000396(n) is located in the square spiral whose vertices are the positive triangular numbers A000217. - Omar E. Pol, May 10 2008

R(a(n)) is prime when R(k) means the digital reverse of k base 2. In base 10, R(a(n)) is prime when R(k) means the digital reverse of k base 10. For example, R(2^53-1) = 1990474529917009 is prime although 2^53-1 is an element of A001348 (not itself prime). - Jonathan Vos Post, Jul 11 2008

Mersenne numbers A000225 whose indices are the prime numbers A000043. - Omar E. Pol, Aug 31 2008

The digital roots are 1 if p == 1 (mod 6) and 4 if p == 5 (mod 6). [T. Koshy, Math Gaz. 89 (2005) p. 465]

Primes p such that for all primes q<p p XOR q = p - q. - Brad Clardy, Oct 26 2011

These primes are Brazilian primes, so they are also in A085104 and A023195. - Bernard Schott, Dec 26 2012

All prime numbers p can be classified by k = (p mod 12) into four classes: k=1, 5, 7, 11. The Mersennne prime numbers 2^p-1, p>2 are in the class k=7 with p=12*(n-1)+7, n=1,2,.... As all 2^p (p odd) are in class k=8 it follows that all 2^p-1, p>2 are in class k=7. - Freimut Marschner, Jul 27 2013

From "The Guinness Book of Primes": "During the reign of Queen Elizabeth I, the largest known prime number was the number of grains of rice on the chess board up to and including the nineteenth square: 524,287 [= 2^19 -1]. By the time Lord Nelson was fighting the Battle of Trafalgar, the record for the largest prime had gone up to the thirty-first square of the chessboard: 2,147,483,647 [= 2^31 -1]. This ten-digits number was proved to be prime in 1772 by the Swiss mathematician Leonard Euler, and it held the record until 1867." [du Sautoy] - Robert G. Wilson v, Nov 26 2013

If n is in the sequence then A024816(n) = antisigma(n) = antisigma(n+1) - 1. Is it true that this sequence gives all numbers n such that antisigma(n) = antisigma(n+1) - 1? Are there composite numbers with this property? - Jaroslav Krizek, Jan 24 2014

If n is in the sequence then phi(n) + sigma(sigma(n)) = 3n.

Is it true that Mersenne primes are all the solutions of the equation phi(x) + sigma(sigma(x)) = 3x? - Farideh Firoozbakht, Sep 03 2014

a(5) = A229381(2) = 8191 is the "Simpsons' Mersenne prime". - Jonathan Sondow, Jan 02 2015

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, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.

Marcus P. F. du Sautoy, The Number Mysteries, A Mathematical Odyssey Through Everyday Life, Palgrave Macmillan, First published in 2010 by the Fourth Estate, an imprint of Harper Collins UK, 2011, p. 46. - Robert G. Wilson v, Nov 26 2013

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.

Harry J. Smith, Table of n, a(n) for n = 1..18

P. Alfeld, The 39th Mersenne prime [From Lekraj Beedassy, Nov 09 2008]

J. Bernheiden, Prime numbers(Prmality check & Mersenne primes: 39th to 43rd)

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.

D. Butler, Mersenne Primes

C. K. Caldwell, Mersenne primes

C. K. Caldwell, "Top Twenty" page, Mersenne Primes

S. A. Khan, Primes in Geometric-Arithmetic Progression, arXiv preprint arXiv:1203.2083, 2012. - From N. J. A. Sloane, Sep 15 2012

Math Reference Project, Mersenne and Fermat Primes

R. Mestrovic, 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, 2012 - From N. J. A. Sloane, Jun 13 2012

L. C. Noll, Mersenne Prime Digits and Names

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

Primefan, The Mersenne Primes

Christian Salas, Cantor Primes as Prime-Valued Cyclotomic Polynomials, arXiv preprint arXiv:1203.3969, 2012.

H. J. Smith, Plot of Mersenne Primes

G. Spence, 36th Mersenne Prime Found

S. Stepney, Mersenne Prime

Thesaurus.maths.org, Mersenne Prime

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

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

Eric Weisstein's World of Mathematics, Mersenne Prime

Eric Weisstein's World of Mathematics, Perfect Number

Eric Weisstein's World of Mathematics, Mersenne Prime

Wikipedia, Mersenne prime

M. Wolf, Computer experiments with Mersenne primes, arXiv preprint arXiv:1112.2412, 2011

a(n) = sigma(A061652(n)) = A000203(A061652(n)). - Omar E. Pol, Apr 15 2008

a(n) = sigma(A019279(n)) = A000203(A019279(n)), provided that there are no odd superperfect numbers. - Omar E. Pol, May 10 2008

a(n) = A000225(A000043(n)). - Omar E. Pol, Aug 31 2008

a(n) = 2^A000043(n) - 1 = 2^(A000005(A061652(n))) - 1. - Omar E. Pol, Oct 27 2011

a(n) = A000040(A059305(n)) = A001348(A016027(n)). - Omar E. Pol, Jun 29 2012

a(n) = A007947(A000396(n))/2, provided that there are no odd perfect numbers. - Omar E. Pol, Feb 01 2013

a(n) = 4*A134709(n) + 3. - Ivan N. Ianakiev, Sep 07 2013

A000668 := proc(n) local i;

i := 2^(ithprime(n))-1:

if (isprime(i)) then

   return i

fi: end:

seq(A000668(n), n=1..31); # - Jani Melik, Feb 09 2011

# Alternate:

seq(numtheory:-mersenne([i]), i=1..26); # Robert Israel, Jul 13 2014

Select[2^Range[1000] - 1, PrimeQ] (* Vladimir Joseph Stephan Orlovsky, Jul 19 2011 *)

(PARI) forprime(p=2, 1e5, if(ispseudoprime(2^p-1), print1(2^p-1", "))) \\ Charles R Greathouse IV, Jul 15 2011

Cf. A000043, A028335 (lengths), A001348, A046051, A057951-A057958, A034876, A124477, A135659, A019279, A061652, A000203, A000217, A000225.

Cf. A085104, A023195, A083420, A023758.

Sequence in context: A136005 A183077 A088552 * A136007 A084732 A123488

Adjacent sequences:  A000665 A000666 A000667 * A000669 A000670 A000671

nonn,nice,hard

N. J. A. Sloane

approved