|
| |
|
|
A091317
|
|
Primes p that divide 2^n+1 for some n.
|
|
4
|
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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, 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, ... , Math. Ann., 168 (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.
|
|
|
FORMULA
|
Has density 17/24 (Hasse).
|
|
|
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
|
| |
|
|
