

A000043


Mersenne exponents: primes p such that 2^p  1 is prime. Then 2^p  1 is called a Mersenne prime.
(Formerly M0672 N0248)


510



2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667
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OFFSET

1,1


COMMENTS

Equivalently, integers n such that 2^n  1 is prime.
It is believed (but unproved) that this sequence is infinite. The data suggest 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^(p1)*M(p) (=A019279*A000668=A000396), has 2p (=A061645) divisors with harmonic mean p (and geometric mean sqrt(N)).  Lekraj Beedassy, 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 nth Mersenne prime (A117293).  Artur Jasinski, Feb 09 2007
Number of divisors of nth even perfect number, divided by 2. Number of divisors of nth even perfect number that are powers of 2. Number of divisors of nth even perfect number that are multiples of nth Mersenne prime A000668(n).  Omar E. Pol, Feb 24 2008
Number of divisors of nth even superperfect number A061652(n). Numbers of divisors of nth superperfect number A019279(n), assuming there are no odd superperfect numbers.  Omar E. Pol, 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)=2512=13 (see A135655, A133033, A090748).  Omar E. Pol, Mar 01 2008
Number of 1's in binary expansion of nth even perfect number (see A135650). Number of 1's in binary expansion of divisors of nth even perfect number that are multiples of nth Mersenne prime A000668(n) (see A135652, A135653, A135654, A135655).  Omar E. Pol, May 04 2008
Indices of the numbers A006516 that are also even perfect numbers.  Omar E. Pol, Aug 30 2008
Indices of Mersenne numbers A000225 that are also Mersenne primes A000668.  Omar E. Pol, Aug 31 2008
A modification of the Eberhart's conjecture proposed by Wagstaff (1983) which proposes that if q_n is the nth prime such that M_(q_n) is a Mersenne prime, then q_n is approximately (2^(e^(gamma)))^n, where gamma is the EulerMascheroni constant. [Weisstein, Wagstaff's Conjecture, see link below]  Jonathan Vos Post, Sep 10 2010
The (prime) number p appears in this sequence if and only if there is no prime q<2^p1 such that the order of 2 modulo q equals p; a special case is that if p=4k+3 is prime and also q=2p+1 is prime then the order of 2 modulo q is p so p is not a term of this sequence.  Joerg Arndt, Jan 16 2011
Primes p such that sigma(2^p)  sigma(2^p1) = 2^p1.  Jaroslav Krizek, Aug 02 2013


REFERENCES

T. M. Apostol, Introduction to Analytic Number Theory, SpringerVerlag, 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.
F. Lemmermeyer, Reciprocity Laws From Euler to Eisenstein, SpringerVerlag, 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 684A15, p. 608.
K. Zsigmondy, Zur Theorie der Potenreste, Monatsh. Math., 3 (1892), 265284.


LINKS

David Wasserman, Table of n, a(n) for n = 1..45 [Updated by N. J. A. Sloane, Feb 06 2013, Alois P. Heinz, May 01 2014, Jan 11 2015, Dec 11 2016]
P. T. Bateman, J. L. Selfridge, S. S. Wagstaff, Jr., The new Mersenne conjecture Amer. Math. Monthly 96 (1989), no. 2, 125128. MR0992073 (90c:11009).
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, arXiv:math/0501118 [math.HO], 20052008.
L. Euler, Observationes de theoremate quodam Fermatiano aliisque ad numeros primos spectantibus
G. Everest et al., Primes generated by recurrence sequences, arXiv:math/0412079 [math.NT], 2006.
G. Everest et al., Primes generated by recurrence sequences, Amer. Math. Monthly, 114 (No. 5, 2007), 417431.
F. Firoozbakht, M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
Donald B. Gillies, Three new Mersenne primes and a statistical theory Mathematics of Computation 18.85 (1964): 9397.
GIMPS (Great Internet Mersenne Prime Search), Distributed Computing Projects
GIMPS, Milestones Report
GIMPS, 48th Known Mersenne Prime Discovered, GIMPS Project Discovers Largest Known Prime Number, 2^57885161  1
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697712. [Annotated scanned copy]
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.
R. Mestrovic, 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
G. P. Michon, Perfect Numbers, Mersenne Primes
Albert A. Mullin, Letter to the editorabout "The new Mersenne conjecture" [Amer. Math. Monthly 96 (1989), no. 2, 125128; MR0992073 (90c:11009)] by P. T. Bateman, J. L. Selfridge and S. S. Wagstaff, Jr., Amer. Math. Monthly 96 (1989), no. 6, 511. MR0999415 (90f:11008).
M. Oakes, A new series of Mersennelike Gaussian primes
Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003.
Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003 [Cached copy, with permission (pdf only)]
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
K. Schneider, PlanetMath.org, Mersenne numbers
H. J. Smith, Mersenne Primes
B. Tuckerman, The 24th Mersenne prime, Proc. Nat. Acad. Sci. USA, 68 (1971), 23192320.
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, Mersenne Prime
Eric Weisstein's World of Mathematics, Cunningham Number
Eric Weisstein's World of Mathematics, Integer Sequence Primes
Eric Weisstein's World of Mathematics, Repunit
Eric Weisstein's World of Mathematics, Mathworld Headline News, 40th Mersenne Prime Announced
Eric Weisstein's World of Mathematics, Mathworld Headline News, 41st Mersenne Prime Announced
Eric Weisstein, MathWorld Headline News, 42nd Mersenne Prime Found
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  Lekraj Beedassy, Sep 18 2008
Eric Weisstein, MathWorld Headline News, 47th Known Mersenne Prime Apparently Discovered  Lekraj Beedassy, Aug 03 2009
Eric W. Weisstein, Wagstaff's Conjecture,  Jonathan Vos Post, Sep 10 2010
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^n1 (or k*2^n+1) is prime
Index entries for "core" sequences


FORMULA

a(n) = log((1/2)*(1+sqrt(1+8*A000396(n))))/log(2).  Artur Jasinski, Sep 23 2008 (under the assumption there are no odd perfect numbers, Joerg Arndt, Feb 23 2014)
a(n) = A000005(A061652(n)).  Omar E. Pol, Aug 26 2009
a(n) = A000120(A000396(n)), assuming there are no odd perfect numbers.  Omar E. Pol, Oct 30 2013
a(n) = 1 + Sum_{m=1..L(n)}(abs(nS(m))abs(nS(m)1/2)+1/2), where S(m) = Sum_{k=1..m}(A010051(k)*A010051(2^k1)) and L(n) >= a(n)1. L(n) can be any function of n which satisfies the inequality.  Timothy Hopper, Jun 11 2015.
a(n) = A260073(A000396(n)) + 1, again assuming there are no odd perfect numbers.  JuriStepan Gerasimov, Aug 29 2015
a(n) = A050475(n)  1.  _Mohammed Bouayoun_, Mar 19 2004


EXAMPLE

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).


MATHEMATICA

Select[ Prime@ Range@ 1000, PrimeQ[2^#  1] &] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 and modified by Robert G. Wilson v, Jan 20 2014 *)
Flatten[Position[EulerPhi[2^#  2] + 2 == EulerPhi[2^#]&/@Range[1, 5000], True]  1] (* Vincenzo Librandi, Aug 31 2015 *)
(* For Mathematica 10.4+ *) Select[Range[10^5], MersennePrimeExponentQ] (* Arkadiusz Wesolowski, Jun 05 2016 *)


PROG

(PARI) isA000043(n) = isprime(2^n1) \\ Michael B. Porter, Oct 28 2009
(PARI)
LL(e)=
{ /* LucasLehmer test for exponent e */
local(n, h);
n = 2^e1;
h = Mod(2, n);
for (k=1, e2, h=2*h*h1);
return( 0==h );
}
forprime(e=2, 5000, if(LL(e), print1(e, ", "))); /* terms<5000, takes 10 secs */
/* Joerg Arndt, Jan 16 2011 */
(PARI) is(n)=my(h=Mod(2, 2^n1)); for(i=1, n2, h=2*h^21); h==0n==2 \\ Charles R Greathouse IV, Jun 05 2013


CROSSREFS

See A000668 for the actual primes, A028335 for their lengths.
Cf. A001348, A016027, A046051, A057429, A057951A057958, A066408, A117293, A127962, A127963, A127964, A127965, A127961, A000979, A000978, A124400, A124401, A127955, A127956, A127957, A127958, A127936, A134458, A000225, A000396, A090748, A133033, A135655, A006516, A019279, A061652, A133033, A135650, A135652, A135653, A135654, A260073, A050475.
Sequence in context: A123856 A120857 A233516 * A109799 A152961 A109461
Adjacent sequences: A000040 A000041 A000042 * A000044 A000045 A000046


KEYWORD

hard,nonn,nice,core,changed


AUTHOR

N. J. A. Sloane


EXTENSIONS

2^6972593  1 is known to be the 38th Mersenne prime.  Harry J. Smith, Apr 17 2003
2^13466917  1 is known to be the 39th Mersenne prime.
Also in the sequence: p=20996011, for which M(p) is a 6.3 million digit number [Nov 17 2003]. Known to be the 40th Mersenne prime since July 2010. See the GIMPS link for details.
Also in the sequence: p=24036583 (for which M(p) is a 7.2 million digit number) [Jun 01 2004]. Known (doublechecked) to be the 41st Mersenne prime since Dec 01 2011.  Jason Kimberley, Jan 05 2012
Also in the sequence: p=25964951 (for which M(p) is a 7.8 million digit number).  Feb 26 2005
Also in the sequence: p=30402457 (for which M(p) is a 9.2 million digit number).  Dec 29 2005
Also in the sequence: p=32582657.  Sep 21 2006
Also in the sequence: p=37156667 and p=43112609.  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, Dec 30 2005
Also in the sequence: p=42643801 (April 2009).
Also in the sequence: p=57885161 (Jan 25 2013).
Added 30402457, now known to be a(43), Joerg Arndt, Mar 04 2014
Added a(44), verified by GIMPS [via Tony Noe], by Charles R Greathouse IV, Nov 10 2014
Also in the sequence: p=74207281.  Charles R Greathouse IV, Jan 19 2016
a(45) added by Heinrich Ludwig, Dec 11 2016


STATUS

approved



