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

Author

N. J. A. Sloane, Feb 21 2004

Keywords

• nonn

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.

Crossrefs

Complement in primes of A014663.

Cf. A014662. - Charles R Greathouse IV, Feb 13 2009

Programs

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 *)

PARI

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

Formula

Has density 17/24 (Hasse 1966).