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A000978 Wagstaff numbers: numbers n such that (2^n + 1)/3 is prime.
(Formerly M2413 N0956)
63
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)
OFFSET

1,1

COMMENTS

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(31) on only give probable primes. Caldwell lists the largest certified primes. - Jens Kruse Andersen, Jan 11 2006

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 (tony.reix(AT)laposte.net), 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. [From Tony Reix (tony.reix(AT)laposte.net), 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

REFERENCES

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.

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

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.

LINKS

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

H. Lifchitz, Mersenne and Fermat primes field

H. & R. Lifchitz, PRP Top Records.

Henri & Renaud Lifchitz, PRP Records.

PRP top list: PRP top [From Tony Reix (tony.reix(AT)laposte.net), Feb 20 2010]

T. Reix, Some Maths about the Vrba-Reix PRP test [From Tony Reix (tony.reix(AT)laposte.net), 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

Yahoo PrimeForm community: PrimeForm [From Tony Reix (tony.reix(AT)laposte.net), Feb 20 2010]

_Ryan Propper_ et al., New Wagstaff PRP exponents, mersenneforum.org

FORMULA

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

MAPLE

Use LLR 3.8.0 by Jean Penne. [From Tony Reix (tony.reix(AT)laposte.net), Feb 20 2010]

MATHEMATICA

a = {}; Do[c = 1 + Sum[2^(2n - 1), {n, 1, x}]; If[PrimeQ[c], AppendTo[a, c]], {x, 0, 100}]; a - Artur Jasinski, Feb 09 2007

PROG

(PARI) forprime(p=2, 1e4, if(ispseudoprime(2^p\/3), print1(p", "))) \\ Charles R Greathouse IV, Jul 15 2011

(Haskell)

a000978 n = a000978_list !! (n-1)

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

-- Reinhard Zumkeller, Mar 24 2013

CROSSREFS

Cf. A107036 = indices of prime Jacobsthal numbers.

Cf. A000979, A124400, A124401, A127955, A127956, A127957, A127958, A127936.

Cf. A010051, A065091, A001045.

Sequence in context: A139758 A060770 A120334 * A128925 A204142 A131261

Adjacent sequences:  A000975 A000976 A000977 * A000979 A000980 A000981

KEYWORD

hard,nonn,nice

AUTHOR

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

EXTENSIONS

a(30) from Kamil Duszenko (kdusz(AT)wp.pl), Feb 03 2003; a(30) was proved prime by Francois Morain with FastECPP. - Tony Reix (tony.reix(AT)laposte.net), 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)=4031399 from Tony Reix (tony.reix(AT)laposte.net), Feb 20 2010

STATUS

approved

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Last modified April 24 19:08 EDT 2014. Contains 240988 sequences.