cp's OEIS Frontend

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

%I A007013 M0866 #134 Feb 16 2025 08:32:31
%S A007013 2,3,7,127,170141183460469231731687303715884105727
%N A007013 Catalan-Mersenne numbers: a(0) = 2; for n >= 0, a(n+1) = 2^a(n) - 1.
%C A007013 The next term is too large to include.
%C A007013 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
%C A007013 Also called the Catalan sequence. - _Artur Jasinski_, Nov 25 2007
%C A007013 a(n) divides a(n+1)-1 for every n. - _Thomas Ordowski_, Apr 03 2016
%C A007013 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
%C A007013 All terms shown are primes, the status of the next term is currently unknown. - _Joerg Arndt_, Apr 03 2016
%C A007013 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
%C A007013 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
%C A007013 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
%C A007013 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
%D A007013 P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 81.
%D A007013 W. Sierpiński, A Selection of Problems in the Theory of Numbers. Macmillan, NY, 1964, p. 91.
%D A007013 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007013 Chris K. Caldwell, <a href="https://t5k.org/mersenne/index.html#c">Mersenne Primes</a>.
%H A007013 Alex Kritov, <a href="https://doi.org/10.13140/RG.2.2.21603.48165/7">Explicit Values for Gravitational and Hubble Constants from Cosmological Entropy Bound and Alpha-Quantization of Particle Masses</a>, 2021, see p. 8.
%H A007013 Double Mersennes Prime Search <a href="http://www.doublemersennes.org/history.php">Status of M(M(p)) where M(p) is a Mersenne prime</a> [outdated link of Will Edgington replaced by _Georg Fischer_, Jan 18 2019].
%H A007013 Carlos Rivera, <a href="https://www.primepuzzles.net/conjectures/conj_015.htm">Conjecture 15. The New Mersenne Conjecture</a>, The Prime Puzzles & Problems Connection.
%H A007013 W. Sierpiński, <a href="/A007013/a007013.pdf">A Selection of Problems in the Theory of Numbers</a>, Macmillan, NY, 1964, p. 91-92. (Annotated scanned copy)
%H A007013 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Catalan-MersenneNumber.html">Catalan-Mersenne Number</a>
%H A007013 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DoubleMersenneNumber.html">Double Mersenne Number</a>.
%F A007013 a(n) = M(a(n-1)) = M^n(2) with M: n-> 2^n-1. - _M. F. Hasler_, Nov 15 2006
%F A007013 A180094(a(n)) = n + 1.
%p A007013 M:=n->2^n-1; '(M@@i)(2)'$i=0..4; # _M. F. Hasler_, Nov 15 2006
%t A007013 NestList[2^#-1&,2,4] (* _Harvey P. Dale_, Jul 18 2011 *)
%o A007013 (PARI) a(n)=if(n,2^a(n-1)-1,2) \\ _Charles R Greathouse IV_, Sep 07 2016
%Y A007013 Cf. A000668, A001567, A014221, A006050, A180094.
%K A007013 nonn
%O A007013 0,1
%A A007013 _N. J. A. Sloane_, Nik Lygeros (webmaster(AT)lygeros.org)
%E A007013 Edited by _Henry Bottomley_, Nov 07 2002
%E A007013 Amended title name by _Marc Morgenegg_, Apr 14 2016