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 A091317 Primes p that divide 2^n+1 for some n. 4

%I

%S 2,3,5,11,13,17,19,29,37,41,43,53,59,61,67,83,97,101,107,109,113,131,

%T 137,139,149,157,163,173,179,181,193,197,211,227,229,241,251,257,269,

%U 277,281,283,293,307,313,317,331,347,349,353,373,379,389,397,401,409,419,421,433

%N Primes p that divide 2^n+1 for some n.

%C From _Charles R Greathouse IV_, Feb 13 2009: (Start)

%C Essentially the same as A014662.

%C Also primes p for which p^2 divides 2^n+1 for some n. If p | 2^g + 1, then 2^g = kp - 1 for some k, so 2^gp = (kp - 1)^p = (-1)^p + (-1)^(p-1) * kp * (p choose 1) + ... and so 2^gp = -1 (mod p^2). (End)

%H T. D. Noe, <a href="/A091317/b091317.txt">Table of n, a(n) for n=1..1000</a>

%H Alexi Block Gorman, Tyler Genao, Heesu Hwang, Noam Kantor, Sarah Parsons, Jeremy Rouse, <a href="http://arxiv.org/abs/1508.02464">The density of primes dividing a particular non-linear recurrence sequence</a>, arXiv:1508.02464 [math.NT], 2015 (see Introduction).

%H H. H. Hasse, <a href="http://resolver.sub.uni-goettingen.de/purl?GDZPPN002295253">Über die Dichte der Primzahlen p, ... </a>, Math. Ann., 168 (1966), 19-23.

%H J. C. Lagarias, <a href="http://projecteuclid.org/euclid.pjm/1102706452">The set of primes dividing the Lucas numbers has density 2/3</a>, Pacific J. Math., 118. No. 2, (1985), 449-461.

%H C. Smyth, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Smyth/smyth2.html">The terms in Lucas Sequences divisible by their indices</a>, JIS 13 (2010) #10.2.4.

%F Has density 17/24 (Hasse).

%p 2, op(select(t -> isprime(t) and numtheory:-order(2,t)::even, [seq(2*i+1, i=1..1000)])); # _Robert Israel_, Aug 12 2015

%t Join[{2}, Select[Prime[Range], EvenQ[MultiplicativeOrder[2, #/ (2^IntegerExponent[#, 2])]]&]] (* _Jean-François Alcover_, Sep 02 2018 *)

%o (PARI) isA091317(p)=!bitand(znorder(Mod(2,p)),1) \\ _Charles R Greathouse IV_, Feb 13 2009

%Y Complement in primes of A014663.

%Y Cf. A014662. - _Charles R Greathouse IV_, Feb 13 2009

%K nonn

%O 1,1

%A _N. J. A. Sloane_, Feb 21 2004

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Last modified October 18 22:26 EDT 2019. Contains 328211 sequences. (Running on oeis4.)