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A007013 Catalan-Mersenne numbers: a(0) = 2; for n >= 0, a(n+1) = 2^a(n) - 1.
(Formerly M0866)
11
2, 3, 7, 127, 170141183460469231731687303715884105727 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

The next term is too large to include.

Orbit of 2 under iteration of the "Mersenne operator" M: n -> 2^n-1 (0 and 1 are fixed points of M). - M. F. Hasler, Nov 15 2006

Called also the Catalan sequence. - Artur Jasinski, Nov 25 2007

a(n) divides a(n+1)-1 for every n. - Thomas Ordowski, Apr 03 2016

Proof: if 2^a == 2 (mod a), then 2^a = 2 + ka for some k, and 2^(2^a-1) = 2^(1 + ka) = 2*(2^a)^k == 2 (mod 2^a-1). Given that a(1) = 3 satisfies 2^a == 2 (mod a), that gives you all 2^a(n) == 2 (mod a(n)), and since a(n+1) - 1 = 2^a(n) - 2 that says a(n) | a(n+1) - 1. - Robert Israel, Apr 05 2016

All terms shown are primes, the status of the next term is currently unknown. - Joerg Arndt, Apr 03 2016

The next term is a prime or a Fermat pseudoprime to base 2 (i.e., a member of A001567). If it is a pseudoprime, then all succeeding terms are pseudoprimes. - Thomas Ordowski, Apr 04 2016

REFERENCES

P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 81.

W. SierpiƄski, A Selection of Problems in the Theory of Numbers. Macmillan, NY, 1964, p. 91.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Table of n, a(n) for n=0..4.

Chris K. Caldwell, Mersenne Primes.

Will Edgington, Status of M(M(p)) where M(p) is a Mersenne prime.

Eric Weisstein's World of Mathematics, Catalan-Mersenne Number

Eric Weisstein's World of Mathematics, Double Mersenne Number.

FORMULA

a(n) = M(a(n-1)) = M^n(2) with M: n-> 2^n-1. - M. F. Hasler, Nov 15 2006

A180094(a(n)) = n + 1.

MAPLE

M:=n->2^n-1; '(M@@i)(2)'$i=0..4; # M. F. Hasler, Nov 15 2006

MATHEMATICA

NestList[2^#-1&, 2, 4] (* Harvey P. Dale, Jul 18 2011 *)

PROG

(PARI) a(n)=if(n, 2^a(n-1)-1, 2) \\ Charles R Greathouse IV, Sep 07 2016

CROSSREFS

Cf. A000668, A001567, A014221.

Sequence in context: A083436 A088856 A173913 * A103405 A087311 A053924

Adjacent sequences:  A007010 A007011 A007012 * A007014 A007015 A007016

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Nik Lygeros (webmaster(AT)lygeros.org)

EXTENSIONS

Edited by Henry Bottomley, Nov 07 2002

Amended title name by Marc Morgenegg, Apr 14 2016

STATUS

approved

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Last modified December 6 06:58 EST 2016. Contains 278775 sequences.