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

2, 3, 7, 127, 170141183460469231731687303715884105727

Offset: 0

Author

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

Keywords

• nonn

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

If the next term were prime, it would be a counterexample to the New Mersenne conjecture. It is known that (2^a(4) + 1) / 3 is composite, with factor 886407410000361345663448535540258622490179142922169401 = 5209834514912200*a(4)+1. - William Hu, Jul 30 2024

a(n) is the smallest number of additive persistence n+1 in base 2. (Similar to A006050 but for binary instead of decimal.) - J. Beach, Nov 17 2024

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

• Chris K. Caldwell, Mersenne Primes.
• Alex Kritov, Explicit Values for Gravitational and Hubble Constants from Cosmological Entropy Bound and Alpha-Quantization of Particle Masses, 2021, see p. 8.
• Double Mersennes Prime Search Status of M(M(p)) where M(p) is a Mersenne prime [outdated link of Will Edgington replaced by _Georg Fischer_, Jan 18 2019].
• Carlos Rivera, Conjecture 15. The New Mersenne Conjecture, The Prime Puzzles & Problems Connection.
• 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.

Crossrefs

Cf. A000668, A001567, A014221, A006050, A180094.

Programs

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 *)

PARI

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

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.

Extensions

Edited by Henry Bottomley, Nov 07 2002

Amended title name by Marc Morgenegg, Apr 14 2016