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A091317
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Primes p that divide 2^n+1 for some n.
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4
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2, 3, 5, 11, 13, 17, 19, 29, 37, 41, 43, 53, 59, 61, 67, 83, 97, 101, 107, 109, 113, 131, 137, 139, 149, 157, 163, 173, 179, 181, 193, 197, 211, 227, 229, 241, 251, 257, 269, 277, 281, 283, 293, 307, 313, 317, 331, 347, 349, 353, 373, 379, 389, 397, 401, 409, 419, 421, 433
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OFFSET
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1,1
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COMMENTS
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From Charles R Greathouse IV, Feb 13 2009: (Start)
Essentially the same as A014662.
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)
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
Alexi Block Gorman, Tyler Genao, Heesu Hwang, Noam Kantor, Sarah Parsons, Jeremy Rouse, The density of primes dividing a particular non-linear recurrence sequence, arXiv:1508.02464 [math.NT], 2015 (see Introduction).
H. H. Hasse, Über die Dichte der Primzahlen p, für die eine vorgegebene ganzrationale Zahl a != 0 von durch eine vorgegebene Primzahl l != 2 teilbarer bzw. unteilbarer Ordnung mod. p ist, Math. Ann., 162 (1965), 74-76.
H. H. Hasse, Über die Dichte der Primzahlen p, für die eine vorgegebene ganzrationale Zahl a != 0 von gerader bzw. ungerader Ordnung mod. p ist, Math. Ann., 166 (1966), 19-23.
J. C. Lagarias, The set of primes dividing the Lucas numbers has density 2/3, Pacific J. Math., 118. No. 2, (1985), 449-461.
C. Smyth, The terms in Lucas Sequences divisible by their indices, JIS 13 (2010) #10.2.4.
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FORMULA
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Has density 17/24 (Hasse 1966).
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MAPLE
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2, op(select(t -> isprime(t) and numtheory:-order(2, t)::even, [seq(2*i+1, i=1..1000)])); # Robert Israel, Aug 12 2015
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MATHEMATICA
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Join[{2}, Select[Prime[Range[100]], EvenQ[MultiplicativeOrder[2, #/ (2^IntegerExponent[#, 2])]]&]] (* Jean-François Alcover, Sep 02 2018 *)
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PROG
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(PARI) isA091317(p)=!bitand(znorder(Mod(2, p)), 1) \\ Charles R Greathouse IV, Feb 13 2009
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CROSSREFS
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Complement in primes of A014663.
Cf. A014662. - Charles R Greathouse IV, Feb 13 2009
Sequence in context: A264866 A038933 A042998 * A088254 A089191 A225184
Adjacent sequences: A091314 A091315 A091316 * A091318 A091319 A091320
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane, Feb 21 2004
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STATUS
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approved
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