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A000978
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Wagstaff numbers: numbers k such that (2^k + 1)/3 is prime.
(Formerly M2413 N0956)
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83
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3, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321, 986191, 4031399
(list;
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OFFSET
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1,1
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COMMENTS
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It is easy to see that the definition implies that k must be an odd prime. - N. J. A. Sloane, Oct 06 2006
The terms from a(32) on only give probable primes as of 2018. Caldwell lists the largest certified primes. - Jens Kruse Andersen, Jan 10 2018
Prime numbers of the form 1+Sum_{i=1..m} 2^(2i-1). - Artur Jasinski, Feb 09 2007
There is a new conjecture stating that a Wagstaff number is prime under the following condition (based on DiGraph cycles under the LLT): Let p be a prime integer > 3, N(p) = 2^p+1 and W(p) = N(p)/3, S(0) = 3/2 (or 1/4) and S(i+1) = S(i)^2 - 2 (mod N(p)). Then W(p) is prime iff S(p-1) == S(0) (mod W(p)). - Tony Reix, Sep 03 2007
As a member of the DUR team (Diepeveen, Underwood, Reix), and thanks to the LLR tool built by Jean Penne, I've found a new and big Wagstaff PRP: (2^4031399+1)/3 is Vrba-Reix PRP! This Wagstaff number has 1,213,572 digits and today is the 3rd biggest PRP ever found. I've done a second verification on a Nehalem core with the PFGW tool. - Tony Reix, Feb 20 2010
13347311 and 13372531 were found to be terms of this sequence (maybe not the next ones) by Ryan Propper in September 2013. - Max Alekseyev, Oct 07 2013
The next term is larger than 10 million. - Gord Palameta, Mar 22 2019
Ryan Propper found another likely term, 15135397, though it only corresponds to a probable prime. - Charles R Greathouse IV, Jul 01 2021
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REFERENCES
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J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
S. S. Wagstaff, Jr., personal communication.
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LINKS
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J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
Eric Weisstein's World of Mathematics, Repunit
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FORMULA
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MATHEMATICA
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Select[Prime[Range[2, 500]], PrimeQ[(2^#+1)/3]&] (* Harvey P. Dale, Jun 13 2022 *)
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PROG
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(Haskell)
a000978 n = a000978_list !! (n-1)
a000978_list = filter ((== 1) . a010051 . a001045) a065091_list
(Python)
from gmpy2 import divexact
from sympy import prime, isprime
A000978 = [p for p in (prime(n) for n in range(2, 10**2)) if isprime(divexact(2**p+1, 3))] # Chai Wah Wu, Sep 04 2014
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CROSSREFS
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Cf. A107036 (indices of prime Jacobsthal numbers).
Cf. A000979, A124400, A124401, A127955, A127956, A127957, A127958, A127936, A010051, A065091, A001045.
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KEYWORD
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nonn,hard,nice,more
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AUTHOR
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EXTENSIONS
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a(30) from Kamil Duszenko (kdusz(AT)wp.pl), Feb 03 2003; a(30) was proved prime by Francois Morain with FastECPP. - Tony Reix, Sep 03 2007
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STATUS
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approved
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