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A000668
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Mersenne primes (of form 2^p - 1 where p is a prime).
(Formerly M2696 N1080)
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299
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3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| See A000043 for the values of p.
Prime repunits in base 2.
Define f(k) = 2k+1; begin with k = 2, a(n+1) = least prime of the form f(f(f(...(a(n)))). - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Dec 26 2003
Mersenne primes other than the first are of form 6n+1. - Lekraj Beedassy (blekraj(AT)yahoo.com), Aug 27 2004
A034876(a(n)) = 0 and A034876(a(n)+1) = 1. - Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Dec 19 2004
Mersenne primes are solutions to sigma(n+1)-sigma(n)=n as perfect numbers (A000396(n)) are solutions to sigma(n)=2n - Benoit Cloitre (benoit7848c(AT)orange.fr), Feb 07 2002
Appears to give all n such that sigma(n+1)-sigma(n)=n - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 27 2002
If n is in the sequence then sigma(sigma(n))=2n+1. Is it true that this sequence gives all numbers n such that sigma(sigma(n))=2n+1? - Farideh Firoozbakht (mymontain(AT)yahoo.com), Aug 19 2005
Mersenne primes other than the first are of form 24n+7; see also A124477 - Artur Jasinski (grafix(AT)csl.pl), Nov 25 2007
It is easily proved that if n is a Mersenne prime then n+sigma(n)=sigma(sigma(n)). Is it true that Mersenne primes are all the solutions of the equation x+sigma(x)=sigma(sigma(x))? - Farideh Firoozbakht (mymontain(AT)yahoo.com), Feb 12 2008
Sum of divisors of n-th even superperfect number A061652(n). Sum of divisors of n-th superperfect number A019279(n), if there are no odd superperfect numbers. - Omar E. Pol (info(AT)polprimos.com), Mar 11 2008
Indices of triangular numbers that are also perfect numbers: A000217(a(n))=A000396(n). - Omar E. Pol (info(AT)polprimos.com), May 10 2008
Number of positive integers (1, 2, 3,...) whose sum is the n-th perfect number A000396(n). - Omar E. Pol (info(AT)polprimos.com), May 10 2008
Vertex number where the n-th perfect number A000396(n) is located in the square spiral whose vertices are the positive triangular numbers A000217. - Omar E. Pol (info(AT)polprimos.com), May 10 2008
R(a(n)) is prime when R(k) means the digital reverse of k base 2. In base 10, R(a(n)) is prime when R(k) means the digital reverse of k base 10. For example, R(2^53-1) = 1990474529917009 is prime although 2^53-1 is an element of A001348 (not itself prime). - Jonathan Vos Post (jvospost3(AT)gmail.com), Jul 11 2008
Mersenne numbers A000225 whose indices are the prime numbers A000043. [From Omar E. Pol (info(AT)polprimos.com), Aug 31 2008]
The digitial roots are 1 if p == 1 (mod 6) and 4 if p == 5 (mod 6). [T. Koshy, Math Gaz. 89 (2005) p. 465]
Primes p such that for all primes q<p p XOR q = p - q. - Brad Clardy, Oct 26 2011
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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.
G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
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.
B. Tuckerman, The 24th Mersenne prime, Proc. Nat. Acad. Sci. USA, 68 (1971), 2319-2320.
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LINKS
| Harry J. Smith, Table of n, a(n) for n=1,...,18
P. Alfeld, The 39th Mersenne prime [From Lekraj Beedassy (blekraj(AT)yahoo.com), Nov 09 2008]
J. Bernheiden, Prime numbers(Prmality check & Mersenne primes:39-th to 43-rd)
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.
D. Butler, Mersenne Primes
C. K. Caldwell, Mersenne primes
C. K. Caldwell, "Top Twenty" page, Mersenne Primes
Math Reference Project, Mersenne and Fermat Primes
L. C. Noll, Mersenne Prime Digits and Names
O. E. Pol, Determinacion geometrica de los numeros primos y perfectos.
Primefan, The Mersenne Primes
H. J. Smith, Plot of Mersenne Primes
G. Spence, 36th Mersenne Prime Found
S. Stepney, Mersenne Prime
Thesaurus.maths.org, Mersenne Prime
B. Tuckerman, The 24th Mersenne Prime
S. S. Wagstaff, Jr., The Cunningham Project
Eric Weisstein's World of Mathematics, Mersenne Prime
Eric Weisstein's World of Mathematics, Perfect Number
Wikipedia, Mersenne prime
Weisstein, Eric W., Mersenne Prime [From Daniel Forgues (squid(AT)zensearch.com), May 16 2010]
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FORMULA
| a(n) = sigma(A061652(n)) = A000203(A061652(n)). - Omar E. Pol, Apr 15 2008
a(n) = sigma(A019279(n)) = A000203(A019279(n)), provided that there are no odd superperfect numbers. - Omar E. Pol, May 10 2008
a(n) = A000225(A000043(n)). - Omar E. Pol, Aug 31 2008
a(n) = 2^A000043(n) - 1 = 2^(A000005(A061652(n))) - 1. - Omar E. Pol, Oct 27 2011
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MAPLE
| A000668 := proc(n) local i;
i := 2^(ithprime(n))-1:
if (isprime(i)) then
RETURN (i)
fi: end:
seq(A000668(n), n=1..31); # - Jani Melik, Feb 09 2011
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MATHEMATICA
| Select[2^Range[1000] - 1, PrimeQ] (* From Vladimir Joseph Stephan Orlovsky, Jul 19 2011 *)
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PROG
| (PARI) forprime(p=2, 1e5, if(ispseudoprime(2^p-1), print1(2^p-1", "))) \\ Charles R Greathouse IV, Jul 15 2011
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CROSSREFS
| Cf. A000043, A028335 (lengths), A001348, A046051, A057951-A057958.
Cf. A034876, A124477, A135659, A019279, A061652, A000203, A000217, A000225.
Sequence in context: A136005 A183077 A088552 * A136007 A084732 A123488
Adjacent sequences: A000665 A000666 A000667 * A000669 A000670 A000671
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KEYWORD
| nonn,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Removed incorrect comment and one trivia comment that does not belong to the OEIS Joerg Arndt (arndt(AT)jjj.de), Mar 11 2010
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