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A007013
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Catalan-Mersenne numbers: a(0) = 2; for n >= 0, a(n+1) = 2^a(n) - 1.
(Formerly M0866)
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17
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OFFSET
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0,1
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COMMENTS
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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
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
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REFERENCES
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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).
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LINKS
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FORMULA
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a(n) = M(a(n-1)) = M^n(2) with M: n-> 2^n-1. - M. F. Hasler, Nov 15 2006
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MAPLE
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M:=n->2^n-1; '(M@@i)(2)'$i=0..4; # M. F. Hasler, Nov 15 2006
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MATHEMATICA
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PROG
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CROSSREFS
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KEYWORD
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nonn,changed
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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