%I A000043 M0672 N0248
%S A000043 2,3,5,7,13,17,19,31,61,89,107,127,521,607,1279,2203,2281,3217,
%T A000043 4253,4423,9689,9941,11213,19937,21701,23209,44497,86243,110503,
%U A000043 132049,216091,756839,859433,1257787,1398269,2976221,3021377,6972593,13466917
%N A000043 Primes p such that 2^p - 1 is prime. 2^p - 1 is then called a Mersenne
prime.
%C A000043 It is believed (but unproved) that this sequence is infinite. The data
suggests 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 (blekraj(AT)yahoo.com), 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 (grafix(AT)csl.pl), 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 (info(AT)polprimos.com),
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 (info(AT)polprimos.com),
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 (info(AT)polprimos.com), Mar 01
2008
%C A000043 Base 2 logarithm of (1 + n-th Mersenne prime A000668(n)). - Omar E. Pol
(info(AT)polprimos.com), Mar 02 2008
%C A000043 Base 2 logarithm of A075398(n). - Omar E. Pol (info(AT)polprimos.com),
Apr 17 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 (info(AT)polprimos.com),
May 04 2008
%C A000043 Indices of the numbers A006516 that are also even perfect numbers. [From
Omar E. Pol (info(AT)polprimos.com), Aug 30 2008]
%C A000043 Indices of Mersenne numbers A000225 that are also Mersenne primes A000668.
[From Omar E. Pol (info(AT)polprimos.com), Aug 31 2008]
%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 G. Everest et al., Primes generated by recurrence sequences, Amer. Math.
Monthly, 114 (No. 5, 2007), 417-431.
%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 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 B. Tuckerman, The 24th Mersenne prime, Notices Amer. Math. Soc., 18 (Jun,
1971), Abstract 684-A15, p. 608.
%D A000043 B. Tuckerman, The 24th Mersenne prime, Proc. Nat. Acad. Sci. USA, 68
(1971), 2319-2320.
%D A000043 K. Zsigmondy, Zur Theorie der Potenreste, Monatsh. Math., 3 (1892), 265-284.
%H A000043 David Wasserman, <a href="b000043.txt">Table of n, a(n) for n = 1..39</
a>
%H A000043 J. Bernheiden, <a href="http://www.mathe-schule.de/download/pdf/Primzahl/
Mersenne.pdf">Mersenne Numbers (Text in German)</a>
%H A000043 Andrew R. Booker, <a href="http://primes.utm.edu/nthprime/">The Nth Prime
Page</a>
%H A000043 J. Brillhart et al., <a href="http://www.ams.org/online_bks/conm22/">
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 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>
%H A000043 L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E026.html">
Observationes do theoremate quodam Fermatiano aliisque ad numeros
primos spectantibus</a>
%H A000043 GIMPS (Great Internet Mersenne Prime Search), <a href="http://www.mersenne.org/
">Distributed Computing Projects</a>
%H A000043 Wilfrid Keller, <a href="http://www.prothsearch.net/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 G. P. Michon, <a href="http://home.att.net/~numericana/answer/numbers.htm#perfect">
Perfect Numbers, Mersenne Primes</a>
%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 O. E. Pol, <a href="http://www.polprimos.com">Determinacion geometrica
de los numeros primos y perfectos</a>.
%H A000043 K. Schneider, PlanetMath.org, <a href="http://planetmath.org/encyclopedia/
MersenneNumbers.html">Mersenne numbers</a>
%H A000043 H. J. Smith, <a href="http://harry-j-smith.com/hjsmithh/Perfect/Mersenne.html">
Mersenne Primes</a>
%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>
%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>
%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, MathWorld Headline News, <a href="http://mathworld.wolfram.com/
news/2009-06-07/mersenne-47/">47th Known Mersenne Prime Apparently
Discovered</a> [From Lekraj Beedassy (blekraj(AT)yahoo.com), Aug
03 2009]
%H A000043 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
MersennePrime.html">Link to a section of The World of Mathematics.</
a>
%H A000043 Eric Weisstein's World of Mathematics, Mathworld Headline News, <a href="http:/
/mathworld.wolfram.com/news/2003-12-02/mersenne">40-th Mersenne Prime
Announced</a>
%H A000043 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
CunninghamNumber.html">Link to a section of The World of Mathematics.</
a>
%H A000043 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
Repunit.html">Link to a section of The World of Mathematics.</a>
%H A000043 Eric Weisstein's World of Mathematics, Mathworld Headline News, <a href="http:/
/mathworld.wolfram.com/news/2004-06-01/mersenne">41st Mersenne Prime
Announced</a>
%H A000043 Eric Weisstein, MathWorld Headline News, <a href="http://mathworld.wolfram.com/
news/2005-02-26/mersenne">42ndMersenne Prime Found</a>
%H A000043 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
IntegerSequencePrimes.html">Integer Sequence Primes</a>
%H A000043 Eric Weisstein, MathWorld Headline News, <a href="http://mathworld.wolfram.com/
news/2005-12-25/mersenne-43">43rd Mersenne Prime Found</a>
%H A000043 Eric Weisstein, MathWorld Headline News, <a href="http://mathworld.wolfram.com/
news/2006-09-11/mersenne-44">44th Mersenne Prime Found</a>
%H A000043 Eric Weisstein, MathWorld Headline News, <a href="http://mathworld.wolfram.com/
news/2008-09-16/mersenne-45-46/">45th and 46th Mersenne Primes Found</
a> [From Lekraj Beedassy (blekraj(AT)yahoo.com), Sep 18 2008]
%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 <a href="Sindx_Pri.html#riesel">Index entries for sequences of n such
that k*2^n-1 (or k*2^n+1) is prime</a>
%F A000043 A000043(n)=Log[(1/2)(1+Sqrt[1+8*A000396(n)])]/Log[2] [From Artur Jasinski
(grafix(AT)csl.pl), Sep 23 2008]
%F A000043 a(n) = A000005(A061652(n)). [From Omar E. Pol (info(AT)polprimos.com),
Aug 26 2009]
%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 ...
%t A000043 Select[Range[10^3],PrimeQ[2^#-1]&] - Vladimir Orlovsky (4vladimir(AT)gmail.com),
Apr 29 2008
%o A000043 (PARI) isA000043(n) = isprime(2^n-1) [From Michael Porter (michael_b_porter(AT)yahoo.com),
Oct 28 2009]
%Y A000043 See A000668 for the actual primes.
%Y A000043 Cf. A001348, A016027, A046051, A057429, A057951-A057958, A066408.
%Y A000043 Cf. also A117293, A127962, A127963, A127964, A127965, A127961, A000979,
A000978, A124400, A124401, A127955, A127956, A127957, A127958, A127936.
%Y A000043 Cf. also A134458, A000225, A000396, A090748, A133033, A135655, A006516,
A019279, A061652, A075398, A133033, A135650, A135652, A135653, A135654.
%Y A000043 Cf. A000005. [From Omar E. Pol (info(AT)polprimos.com), Aug 26 2009]
%Y A000043 Sequence in context: A136003 A123856 A120857 this_sequence A109799 A152961
A109461
%Y A000043 Adjacent sequences: A000040 A000041 A000042 this_sequence A000044 A000045
A000046
%K A000043 hard,nonn,nice,core,new
%O A000043 1,1
%A A000043 N. J. A. Sloane (njas(AT)research.att.com).
%E A000043 2^6972593 - 1 is known to be the 38th Mersenne prime. - Harry J. Smith
(hjsmithh(AT)sbcglobal.net), Apr 17 2003
%E A000043 2^13466917 - 1 is known to be the 39th Mersenne prime.
%E A000043 Also in the sequence: 2^20996011 - 1 (a 6.3 million digit number). -
Nov 17, 2003. See the GIMPS link for details.
%E A000043 Also in the sequence: 2^24036583 - 1 (a 7.2 million digit number). -
Jun 01, 2004
%E A000043 Also in the sequence: 2^25964951 - 1 (a 7.8 million digit number). -
Feb 26, 2005
%E A000043 Also in the sequence: 2^30402457 - 1 (a 9.2 million digit number). -
Dec 29, 2005
%E A000043 Also in the sequence: 2^32582657 - 1. - Sep 21 2006
%E A000043 Also in the sequence: 2^37156667 - 1 and 2^43112609 - 1. - Sep 15 2008
%E A000043 As of Dec 30 2005 the exhaustive search been run through 16693000, according
to the GIMPS status page (thanks to R. K. Guy for this information).
- N. J. A. Sloane (njas(AT)research.att.com), Dec 30 2005
|
