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

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

Offset

1,1

Comments

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)

Links

T. D. Noe, Table of n, a(n) for n=1..1000

Alexi Block Gorman, Tyler Genao, Heesu Hwang, Noam Kantor, Sarah Parsons, and 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.

Eugen J. Ionascu, Florian Luca, and Thomas Merino, On the average value of the minimal Hamming multiple, arXiv:2412.10839 [math.NT], 2024. See pp. 4, 17.

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.

Formula

Has density 17/24 (Hasse 1966).

Maple

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

Mathematica

Join[{2}, Select[Prime[Range[100]], EvenQ[MultiplicativeOrder[2, #/ (2^IntegerExponent[#, 2])]]&]] (* Jean-François Alcover, Sep 02 2018 *)

Prog

(PARI) isA091317(p)=!bitand(znorder(Mod(2, p)), 1) \\ Charles R Greathouse IV, Feb 13 2009

Crossrefs

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

Keyword

nonn

Author

N. J. A. Sloane, Feb 21 2004

Status

approved