%I A000978 M2413 N0956
%S A000978 3,5,7,11,13,17,19,23,31,43,61,79,101,127,167,191,199,313,347,701,1709,
%T A000978 2617,3539,5807,10501,10691,11279,12391,14479,42737,83339,95369,117239,
%U A000978 127031,138937,141079,267017,269987,374321,986191
%N A000978 Wagstaff numbers: numbers n such that (2^n + 1)/3 is prime.
%C A000978 Jun 18 2008: Vincent Diepeveen (diep(AT)xs4all.nl) writes that he has 
               found that (2^986191+1)/3 is a strong PRP, although it may not correspond 
               to the next term. All exponents n=2 up to n=829k have been searched 
               with a single check.
%C A000978 It is easy to see that the definition implies that n must be an odd prime. 
               - N. J. A. Sloane (njas(AT)research.att.com), Oct 06 2006
%C A000978 The terms from a(30)=42737 on only give probable primes. Caldwell lists 
               the largest certified primes. - Jens Kruse Andersen (jens.k.a(AT)get2net.dk), 
               Jan 11 2006
%C A000978 Prime numbers of the form 1+Sum_{i=1..m} [2^(2i-1)]. - Artur Jasinski 
               (grafix(AT)csl.pl), Feb 09 2007
%C A000978 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, Np = 2^p+1 and Wp = N/3, S(0) = 3/2 
               (or 1/4) and S(i+1) = S(i)^2 - 2 (mod Np). Then Wp is prime iff S(p-1) 
               == S(0) (mod Wp) . - Tony Reix (tony.reix(AT)laposte.net), Sep 03 
               2007
%C A000978 a(40) was found by Vincent Diepeveen (see PRP Top Records link). - a(40) 
               from Alexander Adamchuk (alex(AT)kolmogorov.com), Jun 19 2008
%D A000978 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000978 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000978 J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, 
               Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and 
               later supplements.
%D A000978 Problem 174, "A solution in primes", Math. Mag., 27 (1954), 156-157.
%D A000978 S. S. Wagstaff, Jr., personal communication.
%H A000978 J. Brillhart et al., <a href="http://www.ams.org/online_bks/conm22/">
               Factorizations of b^n +- 1</a>, Contemporary Mathematics, Vol. 22, 
               Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
%H A000978 C. Caldwell's The Top Twenty, <a href="http://primes.utm.edu/top20/page.php?id=67">
               Wagstaff</a>.
%H A000978 C. Caldwell, <a href="http://primes.utm.edu/mersenne/NewMersenneConjecture.html">
               New Mersenne Conjecture</a>
%H A000978 H. Dubner and T. Granlund, <a href="http://www.cs.uwaterloo.ca/journals/
               JIS/index.html">Primes of the Form (b^n+1)/(b+1)</a>, J. Integer 
               Sequences, 3 (2000), #P00.2.7.
%H A000978 H. Lifchitz, <a href="http://www.primenumbers.net/Henri/us/MersFermus.htm">
               Mersenne and Fermat primes field</a>
%H A000978 Henri & Renaud Lifchitz, <a href="http://www.primenumbers.net/prptop/
               prptop.php">PRP Records.</a>
%H A000978 S. S. Wagstaff, Jr., <a href="http://www.cerias.purdue.edu/homes/ssw/
               cun/index.html">The Cunningham Project</a>
%H A000978 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Repunit.html">Repunit</a>
%H A000978 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               WagstaffPrime.html">Wagstaff Prime</a>
%H A000978 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               IntegerSequencePrimes.html">Integer Sequence Primes</a>
%H A000978 Wikipedia, <a href="http://en.wikipedia.org/wiki/Wagstaff_prime">Wagstaff 
               prime</a>
%H A000978 H. & R. Lifchitz <a href="http://www.primenumbers.net/prptop/searchform.php?form=%282%5En%2B1%29%2F3&action=S\
               earch">PRP Top Records</a>.
%F A000978 a(n) = A107036(n) for n>1. - Alexander Adamchuk (alex(AT)kolmogorov.com), 
               Feb 10 2007
%t A000978 a = {}; Do[c = 1 + Sum[2^(2n - 1), {n, 1, x}]; If[PrimeQ[c], AppendTo[a, 
               c]], {x, 0, 100}]; a - Artur Jasinski (grafix(AT)csl.pl), Feb 09 
               2007
%Y A000978 Cf. A107036 = indices of prime Jacobsthal numbers.
%Y A000978 Cf. A000979, A124400, A124401, A127955, A127956, A127957, A127958, A127936.
%Y A000978 Sequence in context: A139758 A060770 A120334 this_sequence A128925 A158361 
               A131261
%Y A000978 Adjacent sequences: A000975 A000976 A000977 this_sequence A000979 A000980 
               A000981
%K A000978 hard,nonn,nice
%O A000978 1,1
%A A000978 N. J. A. Sloane (njas(AT)research.att.com), Robert G. Wilson v (rgwv(AT)rgwv.com)
%E A000978 a(30) from Kamil Duszenko (kdusz(AT)wp.pl), Feb 03 2003
%E A000978 a(31) through a(39) from Robert G. Wilson v (rgwv(AT)rgwv.com), Apr 11 
               2005
%E A000978 No other terms below 720000
%E A000978 a(30) has been proved prime by Francois Morain, thanks to FastECPP. - 
               Tony Reix (tony.reix(AT)laposte.net), Sep 03 2007
%E A000978 a(40) from Alexander Adamchuk (alex(AT)kolmogorov.com), Jun 19 2008