

A001122


Primes with primitive root 2.
(Formerly M2473 N0981)


139



3, 5, 11, 13, 19, 29, 37, 53, 59, 61, 67, 83, 101, 107, 131, 139, 149, 163, 173, 179, 181, 197, 211, 227, 269, 293, 317, 347, 349, 373, 379, 389, 419, 421, 443, 461, 467, 491, 509, 523, 541, 547, 557, 563, 587, 613, 619, 653, 659, 661, 677, 701, 709, 757, 773, 787, 797
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OFFSET

1,1


COMMENTS

Artin conjectured that this sequence is infinite.
Conjecture: sequence contains infinitely many pairs of twin primes.  Benoit Cloitre, May 08 2003
Pieter Moree writes (Oct 20 2004): Assuming the Generalized Riemann Hypothesis, it can be shown that the density of primes p such that a prescribed integer g has order (p1)/t, with t fixed, exists and, moreover, it can be computed. This density will be a rational number times the socalled Artin constant. For 2 and 10 the density of primitive roots is A, the Artin constant itself.
It seems that this sequence consists of A050229 \ {1,2}.
Primes p such that 1/p, when written in base 2, has period p1, which is the greatest period possible for any integer.
These are the odd primes p for which the polynomial 1+x+x^2+...+x^(p1) is irreducible over GF(2).  V. Raman, Sep 17 2012 [Corrected by N. J. A. Sloane, Oct 17 2012]
Pollack shows that, on the GRH, that there is some C such that a(n+1)  a(n) < C infinitely often (in fact, 1 can be replaced by any positive integer). Further, for any m, a(n), a(n+1), ..., a(n+m) are consecutive primes infinitely often.  Charles R Greathouse IV, Jan 05 2015
All terms are congruent to 3 or 5 modulo 8. If we define
Pi(N,b) = # {p prime, p <= N, p == b (mod 8)};
Q(N) = # {p prime, p <= N, p in this sequence},
then by Artin's conjecture, Q(N) ~ C*N/log(N) ~ 2*C*(Pi(N,3) + Pi(N,5)), where C = A005596 is Artin's constant.
Conjecture: if we further define
Q(N,b) = # {p prime, p <= N, p == b (mod 8), p in this sequence},
then we have:
Q(N,3) ~ (1/2)*Q(N) ~ C*Pi(N,3);
Q(N,5) ~ (1/2)*Q(N) ~ C*Pi(N,5). (End)


REFERENCES

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 864.
E. Bach and Jeffrey Shallit, Algorithmic Number Theory, I; see p. 221.
J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, New York, 1996; see p. 169.
M. Kraitchik, Recherches sur la Théorie des Nombres. GauthiersVillars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 56.
Lehmer, D. H. and Lehmer, Emma; Heuristics, anyone? in Studies in mathematical analysis and related topics, pp. 202210, Stanford Univ. Press, Stanford, Calif., 1962.
D. Shanks, Solved and Unsolved Problems in Number Theory, 2nd. ed., Chelsea, 1978, p. 81.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].


FORMULA

Delta(a(n),2^a(n)*x) = a(n)*Delta(a(n),2*x), where Delta(k,x) is the difference between numbers of evil(A001969) and odious(A000069) integers divisible by k in interval [0,x).  Vladimir Shevelev, Aug 30 2013


MATHEMATICA

pr = 2; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #  1 &] (* N. J. A. Sloane, Jun 01 2010 *)


PROG

(PARI) forprime(p=3, 1000, if(znorder(Mod(2, p))==(p1), print1(p, ", "))); \\ [corrected by Michel Marcus, Oct 08 2014]
(Python)
from itertools import islice
from sympy import nextprime, is_primitive_root
def A001122_gen(): # generator of terms
p = 2
while (p:=nextprime(p)):
if is_primitive_root(2, p):
yield p


CROSSREFS

Cf. A002326 for the multiplicative order of 2 mod 2n+1. (Alternatively, the least positive value of m such that 2n+1 divides 2^m1).
Cf. A216838 (Odd primes for which 2 is not a primitive root).


KEYWORD

nonn,easy,nice


AUTHOR



STATUS

approved



