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A000668 Mersenne primes (of form 2^p - 1 where p is a prime).
(Formerly M2696 N1080)

%I M2696 N1080

%S 3,7,31,127,8191,131071,524287,2147483647,2305843009213693951,

%T 618970019642690137449562111,162259276829213363391578010288127,

%U 170141183460469231731687303715884105727

%N Mersenne primes (of form 2^p - 1 where p is a prime).

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

%C See A000043 for the values of p.

%C Primes that are repunits in base 2.

%C 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

%C 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

%C A034876(a(n)) = 0 and A034876(a(n)+1) = 1. - _Jonathan Sondow_, Dec 19 2004

%C 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

%C 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

%C 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

%C 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

%C 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

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

%C 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

%C Mersenne numbers A000225 whose indices are the prime numbers A000043. - _Omar E. Pol_, Aug 31 2008

%C 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]

%C Primes p such that for all primes q < p, p XOR q = p - q. - _Brad Clardy_, Oct 26 2011

%C All these primes, except 3, are Brazilian primes, so they are also in A085104 and A023195. - _Bernard Schott_, Dec 26 2012

%C 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

%C 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 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

%C 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

%C 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

%C a(5) = A229381(2) = 8191 is the "Simpsons' Mersenne prime". - _Jonathan Sondow_, Jan 02 2015

%C Equivalently, prime powers of the form 2^n - 1, see Theorem 2 in Lemos & Cambraia Junior. - _Charles R Greathouse IV_, Jul 07 2016

%C 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

%C From _Jaroslav Krizek_, Jan 19 2017: (Start)

%C Primes p such that sigma(p+1) = 2p+1.

%C Primes p such that A051027(p) = sigma(sigma(p)) = 2^k-1 for some k > 1.

%C Primes p of the form sigma(2^prime(n)-1)-1 for some n. Corresponding values of numbers n are in A016027.

%C Primes p of the form sigma(2^(n-1)) for some n > 1. Corresponding values of numbers n are in A000043 (Mersenne exponents).

%C Primes of the form sigma(2^(n+1)) for some n > 1. Corresponding values of numbers n are in A153798 (Mersenne exponents-2).

%C 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).

%C Conjecture: numbers n such that n = sigma(sigma(n+1)-n-1))-1, i.e., A072868(n)-1.

%C Conjecture: primes of the form sigma(4*(n-1)) for some n. Corresponding values of numbers n are in A281312. (End)

%C [Conjecture] For n > 2, the Mersenne number M(n) = 2^n - 1 is a prime if and only if 3^M(n-1) == -1 (mod M(n)). - _Thomas Ordowski_, Aug 12 2018 [This needs proof! - _Joerg Arndt_, Mar 31 2019]

%C Named "Mersenne's numbers" by W. W. Rouse Ball (1892, 1912) after Marin Mersenne (1588-1648). - _Amiram Eldar_, Feb 20 2021

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

%D 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.

%D Graham Everest, Alf van der Poorten, Igor Shparlinski and Thomas Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.

%D 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

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

%H Harry J. Smith, <a href="/A000668/b000668.txt">Table of n, a(n) for n = 1..18</a>

%H Peter Alfeld, <a href="http://www.math.utah.edu/~pa/math/largeprime.html">The 39th Mersenne prime</a>, 2003.

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

%H John Rafael M. Antalan, <a href="https://arxiv.org/abs/1908.06014">A Recreational Application of Two Integer Sequences and the Generalized Repetitious Number Puzzle</a>, arXiv:1908.06014 [math.HO], 2019-2020.

%H W. W. Rouse Ball, <a href="https://catalog.hathitrust.org/Record/000419135">Mathematical recreations and problems of past and present times</a>, London, Macmillan and Co., 1892, pp. 24-25.

%H W. W. Rouse Ball, <a href="https://gdz.sub.uni-goettingen.de/id/PPN599484047_0021">Mersenne's numbers</a>, Messenger of Mathematics, Vol. 21 (1892), pp. 34-40, 121.

%H W. W. Rouse Ball, <a href="https://doi.org/10.1038/089086c0">Mersenne's numbers</a>, Nature, Vol. 89 (1912), p. 86.

%H John Brillhart, D. H. Lehmer, J. L. Selfridge, Bryant Tuckerman and S. S. Wagstaff, Jr., <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 Kevin A. Broughan and Qizhi Zhou, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Broughan/bro32.html">On the Ratio of the Sum of Divisors and Euler's Totient Function II</a>, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.2.

%H Douglas Butler, <a href="https://www.tsm-resources.com/tsm/lists/mers.html">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, "Top Twenty" page, <a href="http://primes.utm.edu/top20/page.php?id=4">Mersenne Primes</a>.

%H Luis H. Gallardo and Olivier Rahavandrainy, <a href="https://arxiv.org/abs/1908.00106">On (unitary) perfect polynomials over F_2 with only Mersenne primes as odd divisors</a>, arXiv:1908.00106 [math.NT], 2019.

%H Richard 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 Sameen Ahmed Khan, <a href="http://arxiv.org/abs/1203.2083">Primes in Geometric-Arithmetic Progression</a>, arXiv preprint arXiv:1203.2083 [math.NT], 2012. - From _N. J. A. Sloane_, Sep 15 2012

%H Abílio Lemos and Ady Cambraia Junior, <a href="https://arxiv.org/abs/1606.08690">On the number of prime factors of Mersenne numbers</a>, arXiv:1606.08690 [math.NT] (2016).

%H Benny Lim, <a href="https://www.parabola.unsw.edu.au/2010-2019/volume-54-2018/issue-3/article/prime-numbers-generated-highly-composite-numbers">Prime Numbers Generated From Highly Composite Numbers</a>, Parabola (2018) Vol. 54, Issue 3.

%H Math Reference Project, <a href="http://www.mathreference.com/num,mers.html">Mersenne and Fermat Primes</a>.

%H Romeo Meštrović, <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. - From _N. J. A. Sloane_, Jun 13 2012

%H Romeo Meštrović, <a href="https://www.researchgate.net/publication/329844912_GOLDBACH-TYPE_CONJECTURES_ARISING_FROM_SOME_ARITHMETIC_PROGRESSIONS">Goldbach-type conjectures arising from some arithmetic progressions</a>, University of Montenegro, 2018.

%H Romeo Meštrović, <a href="https://arxiv.org/abs/1901.07882">Goldbach's like conjectures arising from arithmetic progressions whose first two terms are primes</a>, arXiv:1901.07882 [math.NT], 2019.

%H Landon Curt Noll, <a href="http://www.isthe.com/chongo/tech/math/prime/mersenne.html">Mersenne Prime Digits and Names</a>.

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

%H Primefan, <a href="http://primefan.tripod.com/MersennePrimes.html">The Mersenne Primes</a>.

%H Christian Salas, <a href="http://arxiv.org/abs/1203.3969">Cantor Primes as Prime-Valued Cyclotomic Polynomials</a>, arXiv preprint arXiv:1203.3969 [math.NT], 2012.

%H Harry J. Smith, <a href="http://web.archive.org/web/20120921131527/http://harry-j-smith-memorial.com/Perfect/Mersenne.html">Mersenne Primes</a>, 1981-2010.

%H Gordon Spence, <a href="https://web.archive.org/web/20071221082043/http://www.rugeley.demon.co.uk/gimps/prime.htm">36th Mersenne Prime Found</a>, 1999.

%H Susan Stepney, <a href="https://www-users.cs.york.ac.uk/~susan/cyc/m/mersenne.htm">Mersenne Prime</a>.

%H Thesaurus.maths.org, <a href="http://thesaurus.maths.org/dictionary/map/word/990">Mersenne Prime</a>.

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

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

%H Yunlan Wei, Yanpeng Zheng, Zhaolin Jiang and Sugoog Shon, <a href="https://doi.org/10.3390/math7100893">A Study of Determinants and Inverses for Periodic Tridiagonal Toeplitz Matrices with Perturbed Corners Involving Mersenne Numbers</a>, Mathematics, Vol. 7, No. 10 (2019), 893.

%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/PerfectNumber.html">Perfect Number</a>.

%H Wikipedia, <a href="http://www.wikipedia.org/wiki/Mersenne_prime">Mersenne prime</a>.

%H Marek Wolf, <a href="http://arxiv.org/abs/1112.2412">Computer experiments with Mersenne primes</a>, arXiv preprint arXiv:1112.2412 [math.NT], 2011.

%H Chai Wah Wu, <a href="https://arxiv.org/abs/1805.07431">Can machine learning identify interesting mathematics? An exploration using empirically observed laws</a>, arXiv:1805.07431 [cs.LG], 2018.

%F a(n) = sigma(A061652(n)) = A000203(A061652(n)). - _Omar E. Pol_, Apr 15 2008

%F a(n) = sigma(A019279(n)) = A000203(A019279(n)), provided that there are no odd superperfect numbers. - _Omar E. Pol_, May 10 2008

%F a(n) = A000225(A000043(n)). - _Omar E. Pol_, Aug 31 2008

%F a(n) = 2^A000043(n) - 1 = 2^(A000005(A061652(n))) - 1. - _Omar E. Pol_, Oct 27 2011

%F a(n) = A000040(A059305(n)) = A001348(A016027(n)). - _Omar E. Pol_, Jun 29 2012

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

%F a(n) = 4*A134709(n) + 3. - _Ivan N. Ianakiev_, Sep 07 2013

%F a(n) = A003056(A000396(n)), provided that there are no odd perfect numbers. - _Omar E. Pol_, Dec 19 2016

%F Sum_{n>=1} 1/a(n) = A173898. - _Amiram Eldar_, Feb 20 2021

%p A000668 := proc(n) local i;

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

%p if (isprime(i)) then

%p return i

%p fi: end:

%p seq(A000668(n), n=1..31); # _Jani Melik_, Feb 09 2011

%p # Alternate:

%p seq(numtheory:-mersenne([i]),i=1..26); # _Robert Israel_, Jul 13 2014

%t 2^Array[MersennePrimeExponent, 18] - 1 (* _Jean-François Alcover_, Feb 17 2018, Mersenne primes with less than 1000 digits *)

%o (PARI) forprime(p=2,1e5,if(ispseudoprime(2^p-1),print1(2^p-1", "))) \\ _Charles R Greathouse IV_, Jul 15 2011

%o (PARI) LL(e) = my(n, h); n = 2^e-1; h = Mod(2, n); for (k=1, e-2, h=2*h*h-1); return(0==h) \\ after _Joerg Arndt_ in A000043

%o forprime(p=1, , if(LL(p), print1(p, ", "))) \\ _Felix Fröhlich_, Feb 17 2018

%o (GAP)

%o A000668:=Filtered(List(Filtered([1..600], IsPrime),i->2^i-1),IsPrime); # _Muniru A Asiru_, Oct 01 2017

%o (Python)

%o from sympy import isprime, primerange

%o print([2**n-1 for n in primerange(1, 1001) if isprime(2**n-1)]) # _Karl V. Keller, Jr._, Jul 16 2020

%Y Cf. A000040, A000043, A000203, A000217, A000225, A000396, A001348, A003056, A007947, A016027, A019279, A023195, A023758, A028335 (lengths), A034876, A046051, A057951-A057958, A059305, A061652, A083420, A085104, A124477, A135659, A173898.

%K nonn,nice,hard,changed

%O 1,1

%A _N. J. A. Sloane_

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