This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A000043 M0672 N0248 #557 Jul 17 2025 14:47:08 %S A000043 2,3,5,7,13,17,19,31,61,89,107,127,521,607,1279,2203,2281,3217,4253, %T A000043 4423,9689,9941,11213,19937,21701,23209,44497,86243,110503,132049, %U A000043 216091,756839,859433,1257787,1398269,2976221,3021377,6972593,13466917,20996011,24036583,25964951,30402457,32582657,37156667,42643801,43112609,57885161,74207281 %N A000043 Mersenne exponents: primes p such that 2^p - 1 is prime. Then 2^p - 1 is called a Mersenne prime. %C A000043 Equivalently, integers k such that 2^k - 1 is prime. %C A000043 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 A000043 Length of prime repunits in base 2. %C A000043 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 A000043 In one of his first publications Euler found the numbers up to 31 but erroneously included 41 and 47. %C A000043 Equals number of bits in binary expansion of n-th Mersenne prime (A117293). - _Artur Jasinski_, Feb 09 2007 %C A000043 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 A000043 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 A000043 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 A000043 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 A000043 Indices of the numbers A006516 that are also even perfect numbers. - _Omar E. Pol_, Aug 30 2008 %C A000043 Indices of Mersenne numbers A000225 that are also Mersenne primes A000668. - _Omar E. Pol_, Aug 31 2008 %C A000043 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 A000043 Primes p such that sigma(2^p) - sigma(2^p-1) = 2^p-1. - _Jaroslav Krizek_, Aug 02 2013 %C A000043 Integers k such that every degree k irreducible polynomial over GF(2) is also primitive, i.e., has order 2^k-1. Equivalently, the integers k such that A001037(k) = A011260(k). - _Geoffrey Critzer_, Dec 08 2019 %C A000043 Conjecture: for k > 1, 2^k-1 is (a Mersenne) prime or k = 2^(2^m)+1 (is a Fermat number) if and only if (k-1)^(2^k-2) == 1 (mod (2^k-1)k^2). - _Thomas Ordowski_, Oct 05 2023 %C A000043 Conjecture: for p prime, 2^p-1 is (a Mersenne) prime or p = 2^(2^m)+1 (is a Fermat number) if and only if (p-1)^(2^p-2) == 1 (mod 2^p-1). - _David Barina_, Nov 25 2024 %C A000043 Already as of Dec. 2020, all exponents up to 10^8 had been verified, implying that 74207281, 77232917 and 82589933 are indeed the next three terms. As of today, all exponents up to 130439863 have been tested at least once, see the GIMPS Milestones Report. - _M. F. Hasler_, Apr 11 2025 %C A000043 On June 23. 2025 all exponents up to 74340751 have been verified, confirming that 74207281 is the exponent of the 49th Mersenne Prime. - _Rodolfo Ruiz-Huidobro_, Jun 23 2025 %D A000043 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 4. %D A000043 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 A000043 Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.2 Prime Numbers, p. 79. %D A000043 R. K. Guy, Unsolved Problems in Number Theory, Section A3. %D A000043 F. Lemmermeyer, Reciprocity Laws From Euler to Eisenstein, Springer-Verlag, 2000, p. 57. %D A000043 Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 19. %D A000043 Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 47. %D A000043 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). %D A000043 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %D A000043 James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 132-134. %D A000043 B. Tuckerman, The 24th Mersenne prime, Notices Amer. Math. Soc., 18 (Jun, 1971), Abstract 684-A15, p. 608. %H A000043 P. T. Bateman, J. L. Selfridge, and 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 A000043 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 A000043 Andrew R. Booker, <a href="https://t5k.org/nthprime/">The Nth Prime Page</a> %H A000043 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 A000043 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 A000043 C. K. Caldwell, <a href="http://www.utm.edu/research/primes/mersenne/index.html">Mersenne Primes</a> %H A000043 C. K. Caldwell, <a href="http://www.utm.edu/research/primes/largest.html#largest">Recent Mersenne primes</a> %H A000043 Zuling Chang, Martianus Frederic Ezerman, Adamas Aqsa, Fahreza, San Ling, Janusz Szmidt, and Huaxiong Wang, <a href="https://www.researchgate.net/publication/316819419_Binary_de_Bruijn_Sequences_via_Zech's_Logarithms">Binary de Bruijn Sequences via Zech's Logarithms</a>, 2018. %H A000043 Keith Conrad, <a href="https://kconrad.math.uconn.edu/blurbs/ugradnumthy/squaresandinfmanyprimes.pdf">Square patterns and infinitude of primes</a>, University of Connecticut, 2019. %H A000043 H. Dubner, <a href="/A028491/a028491.pdf">Generalized repunit primes</a>, Math. Comp., 61 (1993), 927-930. [Annotated scanned copy] %H A000043 Leonhard Euler, <a href="https://arxiv.org/abs/math/0501118">Observations on a theorem of Fermat and others on looking at prime numbers</a>, arXiv:math/0501118 [math.HO], 2005-2008. %H A000043 Leonhard Euler, <a href="http://math.dartmouth.edu/~euler/pages/E026.html">Observationes de theoremate quodam Fermatiano aliisque ad numeros primos spectantibus</a> %H A000043 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 A000043 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 A000043 F. Firoozbakht and 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 A000043 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 A000043 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 A000043 GIMPS (Great Internet Mersenne Prime Search), <a href="http://www.mersenne.org/">Distributed Computing Projects</a> %H A000043 GIMPS, <a href="http://www.mersenne.org/report_milestones/">Milestones Report</a> %H A000043 GIMPS, <a href="http://mersenne.org/primes/press/M77232917.html">GIMPS Project discovers largest known prime number 2^77232917-1</a> %H A000043 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 A000043 Wilfrid Keller, <a href="http://www.prothsearch.com/riesel2.html">List of primes k.2^n - 1 for k < 300</a> %H A000043 H. Lifchitz, <a href="http://www.primenumbers.net/Henri/us/MersFermus.htm">Mersenne and Fermat primes field</a> %H A000043 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 A000043 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 A000043 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 A000043 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 A000043 G. P. Michon, <a href="http://www.numericana.com/answer/numbers.htm#perfect">Perfect Numbers, Mersenne Primes</a> %H A000043 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 A000043 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 A000043 M. Oakes, <a href="http://www.mail-archive.com/mersenne@base.com/msg05162.html">A new series of Mersenne-like Gaussian primes</a> %H A000043 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 A000043 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 A000043 Omar E. Pol, <a href="http://www.polprimos.com">Determinacion geometrica de los numeros primos y perfectos</a>. %H A000043 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 A000043 N. J. A. Sloane, <a href="https://arxiv.org/abs/2301.03149">"A Handbook of Integer Sequences" Fifty Years Later</a>, arXiv:2301.03149 [math.NT], 2023, p. 5. %H A000043 H. J. Smith, <a href="http://www.oocities.org/hjsmithh/Perfect/Mersenne.html">Mersenne Primes</a> %H A000043 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 A000043 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 A000043 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 A000043 S. S. Wagstaff, Jr., <a href="http://www.cerias.purdue.edu/homes/ssw/cun/index.html">The Cunningham Project</a> %H A000043 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CunninghamNumber.html">Cunningham Number</a> %H A000043 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IntegerSequencePrimes.html">Integer Sequence Primes</a> %H A000043 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MersennePrime.html">Mersenne Prime</a> %H A000043 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Repunit.html">Repunit</a> %H A000043 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WagstaffsConjecture.html">Wagstaff's Conjecture</a> %H A000043 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 A000043 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 A000043 <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 A000043 <a href="/index/Cor#core">Index entries for "core" sequences</a> %F A000043 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 A000043 a(n) = A000005(A061652(n)). - _Omar E. Pol_, Aug 26 2009 %F A000043 a(n) = A000120(A000396(n)), assuming there are no odd perfect numbers. - _Omar E. Pol_, Oct 30 2013 %e A000043 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 A000043 MersennePrimeExponent[Range[48]] (* _Eric W. Weisstein_, Jul 17 2017; updated Oct 21 2024 *) %o A000043 (PARI) isA000043(n) = isprime(2^n-1) \\ _Michael B. Porter_, Oct 28 2009 %o A000043 (PARI) is(n)=my(h=Mod(2,2^n-1)); for(i=1, n-2, h=2*h^2-1); h==0||n==2 \\ Lucas-Lehmer test for exponent e. - _Joerg Arndt_, Jan 16 2011, and _Charles R Greathouse IV_, Jun 05 2013 %o A000043 forprime(e=2,5000,if(is(e),print1(e,", "))); /* terms < 5000 */ %o A000043 (Python) %o A000043 from sympy import isprime, prime %o A000043 for n in range(1,100): %o A000043 if isprime(2**prime(n)-1): %o A000043 print(prime(n), end=', ') # _Stefano Spezia_, Dec 06 2018 %Y A000043 Cf. A000668 (Mersenne primes). %Y A000043 Cf. A028335 (integer lengths of Mersenne primes). %Y A000043 Cf. A000225 (Mersenne numbers). %Y A000043 Cf. A001348 (Mersenne numbers with n prime). %Y A000043 Cf. 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, A379590. %K A000043 hard,nonn,nice,core %O A000043 1,1 %A A000043 _N. J. A. Sloane_ %E A000043 Also in the sequence: p = 74207281. - _Charles R Greathouse IV_, Jan 19 2016 %E A000043 Also in the sequence: p = 77232917. - _Eric W. Weisstein_, Jan 03 2018 %E A000043 Also in the sequence: p = 82589933. - _Gord Palameta_, Dec 21 2018 %E A000043 a(46) = 42643801 and a(47) = 43112609, whose ordinal positions in the sequence are now confirmed, communicated by _Eric W. Weisstein_, Apr 12 2018 %E A000043 a(48) = 57885161, whose ordinal position in the sequence is now confirmed, communicated by _Benjamin Przybocki_, Jan 05 2022 %E A000043 Also in the sequence: p = 136279841. - _Eric W. Weisstein_, Oct 21 2024 %E A000043 As of Jan 31 2025, 48 terms are known, and are shown in the DATA section. Four additional numbers are known to be in the sequence, namely 74207281, 77232917, 82589933, and 136279841, but they may not be the next terms. See the GIMP website for the latest information. - _N. J. A. Sloane_, Jan 31 2025