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A000978 Wagstaff numbers: numbers n such that (2^n + 1)/3 is prime.
(Formerly M2413 N0956)
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; graph; refs; listen; history; text; internal format)



It is easy to see that the definition implies that n 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 are found to be terms of this sequence (may be 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


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.


Table of n, a(n) for n=1..41.

J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.

C. Caldwell's The Top Twenty, Wagstaff.

C. Caldwell, New Mersenne Conjecture

H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930. [Annotated scanned copy]

H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.

Editor's Note, Table of Wagstaff primes sent by D. H. Lehmer (annotated and scanned copy)

J. E. Foster, Problem 174, A solution in primes, Math. Mag., 27 (1954), 156-157.

R. K. Guy, Letter to N. J. A. Sloane, Aug 1986

R. K. Guy, Letter to N. J. A. Sloane, 1987

H. Lifchitz, Mersenne and Fermat primes field

H. & R. Lifchitz, PRP Top Records.

Henri & Renaud Lifchitz, PRP Records.

Gord Palameta, There are no new Wagstaff primes with exponent below 10 million, mersenneforum.org

Ryan Propper et al., New Wagstaff PRP exponents, mersenneforum.org

PRP top list: PRP top [From Tony Reix, Feb 20 2010]

T. Reix, Some Maths about the Vrba-Reix PRP test [From Tony Reix, Feb 20 2010]

S. S. Wagstaff, Jr., The Cunningham Project

Eric Weisstein's World of Mathematics, Repunit

Eric Weisstein's World of Mathematics, Wagstaff Prime

Eric Weisstein's World of Mathematics, Integer Sequence Primes

Wikipedia, Wagstaff prime

R. G. Wilson, v, Letter to N. J. A. Sloane, circa 1991.

Yahoo PrimeForm community: PrimeForm [From Tony Reix, Feb 20 2010] [Insufficient information to determine which posting to the forum was intended. Probably not worth pursuing. - N. J. A. Sloane, Nov 10 2019]


a(n) = A107036(n) for n>1. - Alexander Adamchuk, Feb 10 2007


Select[Range[5000], PrimeQ[(2^# + 1)/3] &] (* Michael De Vlieger, Jan 10 2018 *)


(PARI) forprime(p=2, 5000, if(ispseudoprime(2^p\/3), print1(p", "))) \\ Charles R Greathouse IV, Jul 15 2011


a000978 n = a000978_list !! (n-1)

a000978_list = filter ((== 1) . a010051 . a001045) a065091_list

-- Reinhard Zumkeller, Mar 24 2013


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


Cf. A107036 (indices of prime Jacobsthal numbers).

Cf. A000979, A124400, A124401, A127955, A127956, A127957, A127958, A127936, A010051, A065091, A001045.

Sequence in context: A246568 A338132 A120334 * A128925 A204142 A131261

Adjacent sequences:  A000975 A000976 A000977 * A000979 A000980 A000981




N. J. A. Sloane, Robert G. Wilson v


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

a(31)-a(39) from Robert G. Wilson v, Apr 11 2005

a(40) from Vincent Diepeveen (diep(AT)xs4all.nl) added by Alexander Adamchuk, Jun 19 2008

a(41) from Tony Reix, Feb 20 2010



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Last modified April 13 22:16 EDT 2021. Contains 342941 sequences. (Running on oeis4.)