

A000668


Mersenne primes (primes of the form 2^n  1).
(Formerly M2696 N1080)


571



3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727
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OFFSET

1,1


COMMENTS

For a Mersenne number 2^n  1 to be prime, the exponent n must itself be prime.
See A000043 for the values of n.
Primes that are 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 the form 6n+1.  Lekraj Beedassy, Aug 27 2004. Mersenne primes other than the first are of the form 24n+7; see also A124477.  Artur Jasinski, Nov 25 2007
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. In fact, 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
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 nth even superperfect number A061652(n). Sum of divisors of nth 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 nth perfect number A000396(n).  Omar E. Pol, May 10 2008
Vertex number where the nth perfect number A000396(n) is located in the square spiral whose vertices are the positive triangular numbers A000217.  Omar E. Pol, May 10 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
All these primes, except 3, 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^p1, p > 2 are in the class k=7 with p=12*(n1)+7, n=1,2,.... As all 2^p (p odd) are in class k=8 it follows that all 2^p1, 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 chessboard 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 thirtyfirst square of the chessboard: 2,147,483,647 [= 2^31  1]. This tendigits 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
Equivalently, prime powers of the form 2^n  1, see Theorem 2 in Lemos & Cambraia Junior.  Charles R Greathouse IV, Jul 07 2016
Primes whose sum of divisors is a power of 2. Primes p such that p + 1 is a power of 2. Primes in A046528.  Omar E. Pol, Jul 09 2016
From Jaroslav Krizek, Jan 19 2017: (Start)
Primes p such that sigma(p+1) = 2p+1.
Primes p such that A051027(p) = sigma(sigma(p)) = 2^k1 for some k > 1.
Primes p of the form sigma(2^prime(n)1)1 for some n. Corresponding values of numbers n are in A016027.
Primes p of the form sigma(2^(n1)) for some n > 1. Corresponding values of numbers n are in A000043 (Mersenne exponents).
Primes of the form sigma(2^(n+1)) for some n > 1. Corresponding values of numbers n are in A153798 (Mersenne exponents2).
Primes p of the form sigma(n) where n is even; subsequence of A023195. Primes p of the form sigma(n) for some n. Conjecture: 31 is the only prime p such that p = sigma(x) = sigma(y) for distinct numbers x and y; 31 = sigma(16) = sigma(25).
Conjecture: numbers n such that n = sigma(sigma(n+1)n1))1, i.e., A072868(n)1.
Conjecture: primes of the form sigma(4*(n1)) for some n. Corresponding values of numbers n are in A281312. (End)
[Conjecture] For n > 2, the Mersenne number M(n) = 2^n  1 is a prime if and only if 3^M(n1) == 1 (mod M(n)).  Thomas Ordowski, Aug 12 2018 [This needs proof!  Joerg Arndt, Mar 31 2019]
Named "Mersenne's numbers" by W. W. Rouse Ball (1892, 1912) after Marin Mersenne (15881648).  Amiram Eldar, Feb 20 2021
Theorem. Let b = 2^p  1 (where p is a prime). Then b is a Mersenne prime iff (c = 2^p  2 is totient or a term of A002202). Otherwise, if c is (nontotient or a term of A005277) then b is composite. Proof. Trivial, since, while b = v^g  1 where v is even, v > 2, g is integer, g > 1, b is always composite, and c = v^g  2 is nontotient (or a term of A005277), and so is for any composite b = 2^g  1 (in the last case, c = v^g  2 is also nontotient, or a term of A005277).  Sergey Pavlov, Aug 30 2021 [Disclaimer: This proof has not been checked.  N. J. A. Sloane, Oct 01 2021]


REFERENCES

Tom M. Apostol, Introduction to Analytic Number Theory, SpringerVerlag, 1976, page 4.
John Brillhart, D. H. Lehmer, J. L. Selfridge, Bryant Tuckerman and S. S. Wagstaff, Jr., Factorizations of b^n + 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
Graham Everest, Alf van der Poorten, Igor Shparlinski and Thomas 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).
Bryant Tuckerman, The 24th Mersenne prime, Notices Amer. Math. Soc., 18 (Jun, 1971), Abstract 684A15, p. 608.


LINKS

Harry J. Smith, Table of n, a(n) for n = 1..18
Peter Alfeld, The 39th Mersenne prime, 2003.
Yan Bingze, Li Qiong, Mao Haokun, and Chen Nan, An efficient hybrid hash based privacy amplification algorithm for quantum key distribution, arXiv:2105.13678 [quantph], 2021.
Andrew R. Booker, The Nth Prime Page
John Rafael M. Antalan, A Recreational Application of Two Integer Sequences and the Generalized Repetitious Number Puzzle, arXiv:1908.06014 [math.HO], 20192020.
W. W. Rouse Ball, Mathematical recreations and problems of past and present times, London, Macmillan and Co., 1892, pp. 2425.
W. W. Rouse Ball, Mersenne's numbers, Messenger of Mathematics, Vol. 21 (1892), pp. 3440, 121.
W. W. Rouse Ball, Mersenne's numbers, Nature, Vol. 89 (1912), p. 86.
John Brillhart, D. H. Lehmer, J. L. Selfridge, Bryant Tuckerman and S. S. Wagstaff, Jr., Factorizations of b^n + 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
Kevin A. Broughan and Qizhi Zhou, On the Ratio of the Sum of Divisors and Euler's Totient Function II, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.2.
Douglas Butler, Mersenne Primes.
C. K. Caldwell, Mersenne primes.
C. K. Caldwell, "Top Twenty" page, Mersenne Primes.
Luis H. Gallardo and Olivier Rahavandrainy, On (unitary) perfect polynomials over F_2 with only Mersenne primes as odd divisors, arXiv:1908.00106 [math.NT], 2019.
Richard K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697712. [Annotated scanned copy]
Sameen Ahmed Khan, Primes in GeometricArithmetic Progression, arXiv preprint arXiv:1203.2083 [math.NT], 2012.  From N. J. A. Sloane, Sep 15 2012
Abílio Lemos and Ady Cambraia Junior, On the number of prime factors of Mersenne numbers, arXiv:1606.08690 [math.NT] (2016).
Benny Lim, Prime Numbers Generated From Highly Composite Numbers, Parabola (2018) Vol. 54, Issue 3.
Math Reference Project, Mersenne and Fermat Primes.
Romeo Meštrović, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012.  From N. J. A. Sloane, Jun 13 2012
Romeo Meštrović, Goldbachtype conjectures arising from some arithmetic progressions, University of Montenegro, 2018.
Romeo Meštrović, Goldbach's like conjectures arising from arithmetic progressions whose first two terms are primes, arXiv:1901.07882 [math.NT], 2019.
Landon Curt 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 PrimeValued Cyclotomic Polynomials, arXiv preprint arXiv:1203.3969 [math.NT], 2012.
Harry J. Smith, Mersenne Primes, 19812010.
Gordon Spence, 36th Mersenne Prime Found, 1999.
Susan Stepney, Mersenne Prime.
Thesaurus.maths.org, Mersenne Prime.
Bryant Tuckerman, The 24th Mersenne prime, Proc. Nat. Acad. Sci. USA, Vol. 68 (1971), pp. 23192320.
Samuel S. Wagstaff, Jr., The Cunningham Project.
Yunlan Wei, Yanpeng Zheng, Zhaolin Jiang and Sugoog Shon, A Study of Determinants and Inverses for Periodic Tridiagonal Toeplitz Matrices with Perturbed Corners Involving Mersenne Numbers, Mathematics, Vol. 7, No. 10 (2019), 893.
Eric Weisstein's World of Mathematics, Mersenne Prime.
Eric Weisstein's World of Mathematics, Perfect Number.
Wikipedia, Mersenne prime.
Marek Wolf, Computer experiments with Mersenne primes, arXiv preprint arXiv:1112.2412 [math.NT], 2011.
Chai Wah Wu, Can machine learning identify interesting mathematics? An exploration using empirically observed laws, arXiv:1805.07431 [cs.LG], 2018.


FORMULA

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
a(n) = A003056(A000396(n)), provided that there are no odd perfect numbers.  Omar E. Pol, Dec 19 2016
Sum_{n>=1} 1/a(n) = A173898.  Amiram Eldar, Feb 20 2021


MAPLE

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


MATHEMATICA

2^Array[MersennePrimeExponent, 18]  1 (* JeanFrançois Alcover, Feb 17 2018, Mersenne primes with less than 1000 digits *)
2^MersennePrimeExponent[Range[18]]  1 (* Eric W. Weisstein, Sep 04 2021 *)


PROG

(PARI) forprime(p=2, 1e5, if(ispseudoprime(2^p1), print1(2^p1", "))) \\ Charles R Greathouse IV, Jul 15 2011
(PARI) LL(e) = my(n, h); n = 2^e1; h = Mod(2, n); for (k=1, e2, h=2*h*h1); return(0==h) \\ after Joerg Arndt in A000043
forprime(p=1, , if(LL(p), print1(p, ", "))) \\ Felix Fröhlich, Feb 17 2018
(GAP)
A000668:=Filtered(List(Filtered([1..600], IsPrime), i>2^i1), IsPrime); # Muniru A Asiru, Oct 01 2017
(Python)
from sympy import isprime, primerange
print([2**n1 for n in primerange(1, 1001) if isprime(2**n1)]) # Karl V. Keller, Jr., Jul 16 2020


CROSSREFS

Cf. A000225 (Mersenne numbers).
Cf. A000043 (Mersenne exponents).
Cf. A001348 (Mersenne numbers with n prime).
Cf. A000040, A000203, A000217, A000396, A003056, A007947, A016027, A019279, A023195, A023758, A028335 (lengths), A034876, A046051, A057951A057958, A059305, A061652, A083420, A085104, A124477, A135659, A173898.
Sequence in context: A336720 A183077 A088552 * A136007 A084732 A123488
Adjacent sequences: A000665 A000666 A000667 * A000669 A000670 A000671


KEYWORD

nonn,nice,hard


AUTHOR

N. J. A. Sloane


STATUS

approved



