OFFSET
1,1
COMMENTS
Equivalently, integers k such that 2^k - 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^(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
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 n-th Mersenne prime (A117293). - Artur Jasinski, Feb 09 2007
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
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
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
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
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
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
Primes p such that sigma(2^p) - sigma(2^p-1) = 2^p-1. - Jaroslav Krizek, Aug 02 2013
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
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
REFERENCES
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 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.
R. K. Guy, Unsolved Problems in Number Theory, Section A3.
F. Lemmermeyer, Reciprocity Laws From Euler to Eisenstein, Springer-Verlag, 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 684-A15, p. 608.
LINKS
David Wasserman, Table of n, a(n) for n = 1..48 [Updated by N. J. A. Sloane, Feb 06 2013, Alois P. Heinz, May 01 2014, Jan 11 2015, Dec 11 2016, Ivan Panchenko, Apr 07 2018, Apr 09 2018, Benjamin Przybocki, Jan 05 2022]
P. T. Bateman, J. L. Selfridge, and S. S. Wagstaff, Jr., The new Mersenne conjecture, Amer. Math. Monthly 96 (1989), no. 2, 125--128. 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.
C. K. Caldwell, Mersenne Primes
C. K. Caldwell, Recent Mersenne primes
Zuling Chang, Martianus Frederic Ezerman, Adamas Aqsa, Fahreza, San Ling, Janusz Szmidt, and Huaxiong Wang, Binary de Bruijn Sequences via Zech's Logarithms, 2018.
Keith Conrad, Square patterns and infinitude of primes, University of Connecticut, 2019.
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930. [Annotated scanned copy]
Leonhard Euler, Observations on a theorem of Fermat and others on looking at prime numbers, arXiv:math/0501118 [math.HO], 2005-2008.
Leonhard 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), 417-431.
F. Firoozbakht and M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
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.
Donald B. Gillies, Three new Mersenne primes and a statistical theory Mathematics of Computation 18.85 (1964): 93-97.
GIMPS (Great Internet Mersenne Prime Search), Distributed Computing Projects
GIMPS, Milestones Report
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [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 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012.
Romeo Meštrović, Goldbach-type 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.
G. P. Michon, Perfect Numbers, Mersenne Primes
Albert A. Mullin, Letter to the editor, 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).
Curt Noll and Laura Nickel, The 25th and 26th Mersenne primes, Math. Comp. 35 (1980), 1387-1390.
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)]
Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 5.
H. J. Smith, Mersenne Primes
B. Tuckerman, The 24th Mersenne prime, Proc. Nat. Acad. Sci. USA, 68 (1971), 2319-2320.
H. S. Uhler, On All Of Mersenne's Numbers Particularly M_193, PNAS 1948 34 (3) 102-103.
H. S. Uhler, First Proof That The Mersenne Number M_157 Is Composite, PNAS 1944 30(10) 314-316.
S. S. Wagstaff, Jr., The Cunningham Project
Eric Weisstein's World of Mathematics, Cunningham Number
Eric Weisstein's World of Mathematics, Integer Sequence Primes
Eric Weisstein's World of Mathematics, Mersenne Prime
Eric Weisstein's World of Mathematics, Repunit
Eric Weisstein's World of Mathematics, Wagstaff's Conjecture
David Whitehouse, Number takes prime position (2^13466917 - 1 found after 13000 years of computer time)
K. Zsigmondy, Zur Theorie der Potenzreste, Monatshefte für Mathematik und Physik, Vol. 3, No. 1 (1892), 265-284.
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) = 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
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
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
MersennePrimeExponent[Range[47]] (* Eric W. Weisstein, Jul 17 2017 *)
PROG
(PARI) isA000043(n) = isprime(2^n-1) \\ Michael B. Porter, Oct 28 2009
(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
forprime(e=2, 5000, if(is(e), print1(e, ", "))); /* terms < 5000 */
(Python)
from sympy import isprime, prime
for n in range(1, 100):
if isprime(2**prime(n)-1):
print(prime(n), end=', ') # Stefano Spezia, Dec 06 2018
CROSSREFS
Cf. A000668 (Mersenne primes).
Cf. A028335 (integer lengths of Mersenne primes).
Cf. A000225 (Mersenne numbers).
Cf. A001348 (Mersenne numbers with n prime).
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.
KEYWORD
hard,nonn,nice,core,changed
AUTHOR
EXTENSIONS
Also in the sequence: p = 74207281. - Charles R Greathouse IV, Jan 19 2016
Also in the sequence: p = 77232917. - Eric W. Weisstein, Jan 03 2018
Also in the sequence: p = 82589933. - Gord Palameta, Dec 21 2018
a(46) = 42643801 and a(47) = 43112609, whose ordinal positions in the sequence are now confirmed, communicated by Eric W. Weisstein, Apr 12 2018
a(48) = 57885161, whose ordinal position in the sequence is now confirmed, communicated by Benjamin Przybocki, Jan 05 2022
STATUS
approved