Search: id:A000043 Results 1-1 of 1 results found. %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, Table of n, a(n) for n = 1..39 %H A000043 J. Bernheiden, Mersenne Numbers (Text in German) %H A000043 Andrew R. Booker, The Nth Prime Page %H A000043 J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002. %H A000043 P. G. Brown, A Note on Ramanujan's (FALSE) Conjectures Regarding 'Mersenne Primes' %H A000043 C. K. Caldwell, Mersenne Primes %H A000043 C. K. Caldwell, Recent Mersenne primes %H A000043 L. Euler, Observations on a theorem of Fermat and others on looking at prime numbers %H A000043 L. Euler, Observationes do theoremate quodam Fermatiano aliisque ad numeros primos spectantibus %H A000043 GIMPS (Great Internet Mersenne Prime Search), Distributed Computing Projects %H A000043 Wilfrid Keller, List of primes k.2^n - 1 for k < 300 %H A000043 H. Lifchitz, Mersenne and Fermat primes field %H A000043 A. J. Menezes, P. C. van Oorschot and S. A. Vanstone, Handbook of Applied Cryptography, CRC Press, 1996; see p. 143. %H A000043 G. P. Michon, Perfect Numbers, Mersenne Primes %H A000043 M. Oakes, A new series of Mersenne-like Gaussian primes %H A000043 O. E. Pol, Determinacion geometrica de los numeros primos y perfectos. %H A000043 K. Schneider, PlanetMath.org, Mersenne numbers %H A000043 H. J. Smith, Mersenne Primes %H A000043 H. S. Uhler, On All Of Mersenne's Numbers Particularly M_193 %H A000043 H. S. Uhler, First Proof That The Mersenne Number M_157 Is Composite %H A000043 S. S. Wagstaff, Jr., The Cunningham Project %H A000043 Eric Weisstein, MathWorld Headline News, 47th Known Mersenne Prime Apparently Discovered [From Lekraj Beedassy (blekraj(AT)yahoo.com), Aug 03 2009] %H A000043 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A000043 Eric Weisstein's World of Mathematics, Mathworld Headline News, 40-th Mersenne Prime Announced %H A000043 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A000043 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A000043 Eric Weisstein's World of Mathematics, Mathworld Headline News, 41st Mersenne Prime Announced %H A000043 Eric Weisstein, MathWorld Headline News, 42ndMersenne Prime Found %H A000043 Eric Weisstein's World of Mathematics, Integer Sequence Primes %H A000043 Eric Weisstein, MathWorld Headline News, 43rd Mersenne Prime Found %H A000043 Eric Weisstein, MathWorld Headline News, 44th Mersenne Prime Found %H A000043 Eric Weisstein, MathWorld Headline News, 45th and 46th Mersenne Primes Found [From Lekraj Beedassy (blekraj(AT)yahoo.com), Sep 18 2008] %H A000043 David Whitehouse, Number takes prime position (2^13466917 - 1 found after 13000 years of computer time) %H A000043 Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime %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 Search completed in 0.004 seconds