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A000043 Mersenne exponents: primes p such that 2^p - 1 is prime. Then 2^p - 1 is called a Mersenne prime.
(Formerly M0672 N0248)

%I M0672 N0248

%S 2,3,5,7,13,17,19,31,61,89,107,127,521,607,1279,2203,2281,3217,4253,

%T 4423,9689,9941,11213,19937,21701,23209,44497,86243,110503,132049,

%U 216091,756839,859433,1257787,1398269,2976221,3021377,6972593,13466917,20996011,24036583,25964951,30402457,32582657,37156667

%N Mersenne exponents: primes p such that 2^p - 1 is prime. Then 2^p - 1 is called a Mersenne prime.

%C Equivalently, integers n such that 2^n - 1 is prime.

%C 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.

%C Length of prime repunits in base 2.

%C 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_, Aug 21 2004

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

%C Equals number of bits in binary expansion of n-th Mersenne prime (A117293). - _Artur Jasinski_, Feb 09 2007

%C 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_, Feb 24 2008

%C 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_, Mar 01 2008

%C 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_, Mar 01 2008

%C 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_, May 04 2008

%C Indices of the numbers A006516 that are also even perfect numbers. - _Omar E. Pol_, Aug 30 2008

%C Indices of Mersenne numbers A000225 that are also Mersenne primes A000668. - _Omar E. Pol_, Aug 31 2008

%C The (prime) number p appears in this sequence if and only if there is no prime q<2^p-1 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

%C Primes p such that sigma(2^p) - sigma(2^p-1) = 2^p-1. - _Jaroslav Krizek_, Aug 02 2013

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

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

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

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

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

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

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

%H David Wasserman, <a href="/A000043/b000043.txt">Table of n, a(n) for n = 1..45</a> [Updated by _N. J. A. Sloane_, Feb 06 2013, _Alois P. Heinz_, May 01 2014, Jan 11 2015, Dec 11 2016]

%H P. T. Bateman, J. L. Selfridge, S. S. Wagstaff, Jr., <a href="http://www.jstor.org/stable/2323195">The new Mersenne conjecture</a> Amer. Math. Monthly 96 (1989), no. 2, 125--128. MR0992073 (90c:11009).

%H J. Bernheiden, <a href="http://web.archive.org/web/20160412214003/http://www.mathe-schule.de/download/pdf/Primzahl/Mersenne.pdf">Mersenne Numbers (Text in German)</a>

%H Andrew R. Booker, <a href="http://primes.utm.edu/nthprime/">The Nth Prime Page</a>

%H J. Brillhart et al., <a href="http://dx.doi.org/10.1090/conm/022">Factorizations of b^n +- 1</a>, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.

%H P. G. Brown, <a href="http://www.austms.org.au/Publ/Gazette/1997/Nov97/brown.html">A Note on Ramanujan's (FALSE) Conjectures Regarding 'Mersenne Primes'</a>

%H C. K. Caldwell, <a href="http://www.utm.edu/research/primes/mersenne/index.html">Mersenne Primes</a>

%H C. K. Caldwell, <a href="http://www.utm.edu/research/primes/largest.html#largest">Recent Mersenne primes</a>

%H H. Dubner, <a href="/A028491/a028491.pdf">Generalized repunit primes</a>, Math. Comp., 61 (1993), 927-930. [Annotated scanned copy]

%H L. Euler, <a href="http://arXiv.org/abs/math.HO/0501118">Observations on a theorem of Fermat and others on looking at prime numbers</a>, arXiv:math/0501118 [math.HO], 2005-2008.

%H L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E026.html">Observationes de theoremate quodam Fermatiano aliisque ad numeros primos spectantibus</a>

%H G. Everest et al., <a href="http://arxiv.org/abs/math/0412079">Primes generated by recurrence sequences</a>, arXiv:math/0412079 [math.NT], 2006.

%H G. Everest et al., <a href="http://www.jstor.org/stable/27642221">Primes generated by recurrence sequences</a>, Amer. Math. Monthly, 114 (No. 5, 2007), 417-431.

%H F. Firoozbakht, M. F. Hasler, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Hasler/hasler2.html">Variations on Euclid's formula for Perfect Numbers</a>, JIS 13 (2010) #10.3.1.

%H Donald B. Gillies, <a href="http://dx.doi.org/10.1090/S0025-5718-1964-0159774-6">Three new Mersenne primes and a statistical theory</a> Mathematics of Computation 18.85 (1964): 93-97.

%H GIMPS (Great Internet Mersenne Prime Search), <a href="http://www.mersenne.org/">Distributed Computing Projects</a>

%H GIMPS, <a href="http://www.mersenne.org/report_milestones/">Milestones Report</a>

%H GIMPS, <a href="http://mersenne.org/primes/press/M77232917.html">GIMPS Project discovers largest known prime number 2^77232917-1</a>

%H R. K. Guy, <a href="/A005165/a005165.pdf">The strong law of small numbers</a>. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]

%H Wilfrid Keller, <a href="http://web.archive.org/web/20161028021640/http://www.prothsearch.net/riesel2.html">List of primes k.2^n - 1 for k < 300</a>

%H H. Lifchitz, <a href="http://www.primenumbers.net/Henri/us/MersFermus.htm">Mersenne and Fermat primes field</a>

%H A. J. Menezes, P. C. van Oorschot and S. A. Vanstone, <a href="http://www.cacr.math.uwaterloo.ca/hac/">Handbook of Applied Cryptography</a>, CRC Press, 1996; see p. 143.

%H R. Mestrovic, <a href="http://arxiv.org/abs/1202.3670">Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof</a>, arXiv preprint arXiv:1202.3670 [math.HO], 2012.

%H G. P. Michon, <a href="http://www.numericana.com/answer/numbers.htm#perfect">Perfect Numbers, Mersenne Primes</a>

%H Albert A. Mullin, <a href="http://www.jstor.org/stable/2323972">Letter to the editor</a>, about "The new Mersenne conjecture" [Amer. Math. Monthly 96 (1989), no. 2, 125-128; 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).

%H Curt Noll and Laura Nickel, <a href="https://doi.org/10.1090/S0025-5718-1980-0583517-4">The 25th and 26th Mersenne primes</a>, Math. Comp. 35 (1980), 1387-1390.

%H M. Oakes, <a href="http://www.mail-archive.com/mersenne@base.com/msg05162.html">A new series of Mersenne-like Gaussian primes</a>

%H Ed Pegg, Jr., <a href="http://www.mathpuzzle.com/MAA/07-Sequence%20Pictures/mathgames_12_08_03.html">Sequence Pictures</a>, Math Games column, Dec 08 2003.

%H Ed Pegg, Jr., <a href="/A000043/a000043_2.pdf">Sequence Pictures</a>, Math Games column, Dec 08 2003 [Cached copy, with permission (pdf only)]

%H Omar E. Pol, <a href="http://www.polprimos.com">Determinacion geometrica de los numeros primos y perfectos</a>.

%H Maxie D. Schmidt, <a href="https://arxiv.org/abs/1701.04741">New Congruences and Finite Difference Equations for Generalized Factorial Functions</a>, arXiv:1701.04741 [math.CO], 2017.

%H H. J. Smith, <a href="http://www.oocities.org/hjsmithh/Perfect/Mersenne.html">Mersenne Primes</a>

%H B. Tuckerman, <a href="http://www.pnas.org/content/68/10/2319.abstract">The 24th Mersenne prime</a>, Proc. Nat. Acad. Sci. USA, 68 (1971), 2319-2320.

%H H. S. Uhler, <a href="http://www.pnas.org/cgi/reprint/34/3/102.pdf">On All Of Mersenne's Numbers Particularly M_193</a>, PNAS 1948 34 (3) 102-103.

%H H. S. Uhler, <a href="http://www.pnas.org/cgi/reprint/30/10/314.pdf">First Proof That The Mersenne Number M_157 Is Composite</a>, PNAS 1944 30(10) 314-316.

%H S. S. Wagstaff, Jr., <a href="http://www.cerias.purdue.edu/homes/ssw/cun/index.html">The Cunningham Project</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CunninghamNumber.html">Cunningham Number</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IntegerSequencePrimes.html">Integer Sequence Primes</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MersennePrime.html">Mersenne Prime</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Repunit.html">Repunit</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WagstaffsConjecture.html"> Wagstaff's Conjecture</a>

%H David Whitehouse, <a href="http://news.bbc.co.uk/hi/english/sci/tech/newsid_1693000/1693364.stm">Number takes prime position</a> (2^13466917 - 1 found after 13000 years of computer time)

%H K. Zsigmondy, <a href="https://doi.org/10.1007/BF01692444">Zur Theorie der Potenzreste</a>, Monatshefte für Mathematik und Physik, Vol. 3, No. 1 (1892), 265-284.

%H <a href="/index/Pri#riesel">Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime</a>

%H <a href="/index/Cor#core">Index entries for "core" sequences</a>

%F 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)

%F a(n) = A000005(A061652(n)). - _Omar E. Pol_, Aug 26 2009

%F a(n) = A000120(A000396(n)), assuming there are no odd perfect numbers. - _Omar E. Pol_, Oct 30 2013

%F a(n) = 1 + Sum_{m=1..L(n)}(abs(n-S(m))-abs(n-S(m)-1/2)+1/2), where S(m) = Sum_{k=1..m}(A010051(k)*A010051(2^k-1)) and L(n) >= a(n)-1. L(n) can be any function of n which satisfies the inequality. - _Timothy Hopper_, Jun 11 2015.

%F a(n) = A260073(A000396(n)) + 1, again assuming there are no odd perfect numbers. Also, a(n) = A050475(n) - 1. - _Juri-Stepan Gerasimov_, Aug 29 2015

%e 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).

%t MersennePrimeExponent[Range[45]] (* _Eric W. Weisstein_, Jul 17 2017 *)

%o (PARI) isA000043(n) = isprime(2^n-1) \\ _Michael B. Porter_, Oct 28 2009

%o (PARI)

%o LL(e)=

%o { /* Lucas-Lehmer test for exponent e */

%o my(n, h);

%o n = 2^e-1;

%o h = Mod(2,n);

%o for (k=1, e-2, h=2*h*h-1);

%o return( 0==h );

%o }

%o forprime(e=2,5000,if(LL(e),print1(e,", "))); /* terms<5000, takes 10 secs */

%o /* _Joerg Arndt_, Jan 16 2011 */

%o (PARI) is(n)=my(h=Mod(2,2^n-1)); for(i=1, n-2, h=2*h^2-1); h==0||n==2 \\ _Charles R Greathouse IV_, Jun 05 2013

%Y See A000668 for the actual primes, A028335 for their lengths.

%Y Cf. A001348, A016027, A046051, A057429, A057951-A057958, 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.

%K hard,nonn,nice,core,changed

%O 1,1

%A _N. J. A. Sloane_

%E As of January 3, 2018, GIMPS participants have tested and verified all exponents below 42033653, and tested all exponents below 76333099 at least once. - _Eric W. Weisstein_, Jan 03 2018

%E Also in the sequence: p = 42643801 (Jun 12 2009).

%E Also in the sequence: p = 43112609 (Aug 23 2008).

%E Also in the sequence: p = 57885161 (Jan 25 2013).

%E Also in the sequence: p = 74207281. - _Charles R Greathouse IV_, Jan 19 2016

%E Also in the sequence: p = 77232917. - _Eric W. Weisstein_, Jan 03 2018

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