

A000043


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


673



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, 42643801, 43112609, 57885161
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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.
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
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
Integers k such that every degree k irreducible polynomial over GF(2) is also primitive, i.e., has order 2^k1. Equivalently, the integers k such that A001037(k) = A011260(k).  Geoffrey Critzer, Dec 08 2019
Conjecture: for k > 1, 2^k1 is (a Mersenne) prime or k = 2^(2^m)+1 (is a Fermat number) if and only if (k1)^(2^k2) == 1 (mod (2^k1)k^2).  Thomas Ordowski, Oct 05 2023


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.


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, 125128. MR0992073 (90c:11009).
J. Brillhart et al., Factorizations of b^n + 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
Albert A. Mullin, Letter to the editor, about "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).
Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003 [Cached copy, with permission (pdf only)]
Eric Weisstein's World of Mathematics, Repunit


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(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


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



PROG

(PARI) is(n)=my(h=Mod(2, 2^n1)); for(i=1, n2, h=2*h^21); h==0n==2 \\ LucasLehmer 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):


CROSSREFS

Cf. A028335 (integer lengths of Mersenne primes).
Cf. A001348 (Mersenne numbers with n prime).
Cf. 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.


KEYWORD

hard,nonn,nice,core


AUTHOR



EXTENSIONS

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



