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A007013 Catalan-Mersenne numbers: a(0) = 2; for n >= 0, a(n+1) = 2^a(n) - 1.
(Formerly M0866)
14
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

Also called 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

a(n) is the least positive integer that requires n+1 steps to reach 1 under iteration of the binary weight function A000120. - David Radcliffe, Jun 25 2018

REFERENCES

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

W. Sierpiński, 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.

Double Mersennes Prime Search Status of M(M(p)) where M(p) is a Mersenne prime [outdated link of Will Edgington replaced by Georg Fischer, Jan 18 2019].

W. Sierpiński, A Selection of Problems in the Theory of Numbers, Macmillan, NY, 1964, p. 91-92. (Annotated scanned copy)

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 March 26 23:02 EDT 2019. Contains 321565 sequences. (Running on oeis4.)