A054639 -id:A054639 - cp's OEIS frontend

A071642 Numbers n such that x^n + x^(n-1) + x^(n-2) + ... + x + 1 is irreducible over GF(2).

0, 1, 2, 4, 10, 12, 18, 28, 36, 52, 58, 60, 66, 82, 100, 106, 130, 138, 148, 162, 172, 178, 180, 196, 210, 226, 268, 292, 316, 346, 348, 372, 378, 388, 418, 420, 442, 460, 466, 490, 508, 522, 540, 546, 556, 562, 586, 612, 618, 652, 658, 660, 676, 700, 708, 756, 772

Offset: 1

N. J. A. Sloane, Jun 22 2002

All such polynomials of odd degree > 1 are reducible over GF(2).
For n >= 2, a(n) = A001122(n-2) - 1 due to the relationship between cycles and irreducibility. - T. D. Noe, Sep 09 2003
n such that a type-1 optimal normal basis of GF(2^n) (over GF(2)) exists. The corresponding field polynomial is the all-ones polynomial x^n+x^(n-1)+...+1. - Joerg Arndt, Feb 25 2008
From Peter R. J. Asveld, Aug 13 2009: (Start)
a(n) is also the n-th S-prime (Shuffle prime)
For N>=2, the family of shuffle permutations is defined by
p(m,N) = 2m (mod N+1) if N is even,
p(m,N) = 2m (mod N) if N is odd and 1<=mp(N,N) = N if N is odd.
N is S-prime if p(m,N) consists of a single cycle of length N.
So all S-primes are even.
N is S-prime iff p=N+1 is an odd prime number and +2 generates Z_p^* (the multiplicative group of Z_p).
a(n)/2 results in the Josephus_2-primes (A163782). Considered as sets a(n)/2 is the union of A163777 and A163779. If b(n) denotes the dual shuffle primes (A163776), then the union of a(n)/2 and b(n)/2 is equal to the Twist-primes or Queneau numbers (A054639); their intersection is equal to the Archimedes_0-primes (A163777). (End)
Conjecture: Terms >= 2 are numbers n such that P^n + P^(n-1) + P^(n-2) + ... + P + 1 is irreducible over GF(2), where P=x^2+x+1. - Luis H. Gallardo, Dec 23 2019

			For n=4 and n=6 we obtain the permutations (1 2 4 3) and (1 2 4)(3 6 5): 4 is S-prime, but 6 is not. [_Peter R. J. Asveld_, Aug 13 2009]

P. R. J. Asveld, Table of n, a(n) for n = 1..3605
Joerg Arndt, Matters Computational (The Fxtbook), section 42.9 "Gaussian normal bases", pp.914-920
P. R. J. Asveld, Permuting operations on strings and their relation to prime numbers, Discrete Applied Mathematics 159 (2011) 1915-1932.
P. R. J. Asveld, Permuting operations on strings and the distribution of their prime numbers, (2011), TR-CTIT-11-24, Dept. of CS, Twente University of Technology, Enschede, The Netherlands.
P. R. J. Asveld, Some Families of Permutations and Their Primes (2009), TR-CTIT-09-27, Dept. of CS, Twente University of Technology, Enschede, The Netherlands.
P. R. J. Asveld, Permuting Operations on Strings-Their Permutations and Their Primes, Twente University of Technology, 2014.
Hélianthe Caure, Canons rythmiques et pavages modulaires, Thesis Université Pierre et Marie Curie - Paris VI, 2016. See page 108. In French.
H. Caure, C. Agon, and M. Andreatta, Modulus p Rhythmic Tiling Canons and some implementations in Open Music visual programming language, in Proceedings ICMC|SMC|2014 14-20 September 2014, Athens, Greece.
M. Olofsson, VLSI Aspects on Inversion in Finite Fields, Dissertation No. 731, Dept. Elect. Engin., Linkoping, Sweden, 2002.
Eric Weisstein's World of Mathematics, Irreducible Polynomial
Index entries for sequences related to the Josephus Problem

Cf. A001122 (primes with primitive root 2).

Join[{0, 1}, Reap[For[p = 2, p < 10^3, p = NextPrime[p], If[ MultiplicativeOrder[2, p] == p-1, Sow[p-1]]]][[2, 1]]] (* Jean-François Alcover, Dec 10 2015, adapted from PARI *)

forprime(p=3,1000,if(znorder(Mod(2,p))==p-1,print1(p-1,", "))) /* Joerg Arndt, Jul 05 2011 */

Extended by Robert G. Wilson v, Jun 24 2002

Initial terms of b-file corrected by N. J. A. Sloane, Aug 31 2009

A216371 Odd primes with one coach: primes p such that A135303((p-1)/2) = 1.

Original entry on oeis.org

3, 5, 7, 11, 13, 19, 23, 29, 37, 47, 53, 59, 61, 67, 71, 79, 83, 101, 103, 107, 131, 139, 149, 163, 167, 173, 179, 181, 191, 197, 199, 211, 227, 239, 263, 269, 271, 293, 311, 317, 347, 349, 359, 367, 373, 379, 383, 389, 419, 421, 443, 461, 463, 467, 479, 487

Offset: 1

Gary W. Adamson, Sep 05 2012

Given that prime p has only one coach, the corresponding value of k in A003558 must be (p-1)/2, and vice versa. Using the Coach theorem of Jean Pedersen et al., phi(b) = 2 * c * k, with b odd. Let b = p, prime. Then phi(p) = (p-1), and k must be (p-1)/2 iff c = 1. Or, phi(p) = (p-1) = 2 * 1 * (p-1)/2.
Conjecture relating to odd integers: iff an integer is in the set A216371 and is either of the form 4q - 1 or 4q + 1, (q>0); then the top row of its coach (cf. A003558) is composed of a permutation of the first q odd integers. Examples: 11 is of the form 4q - 1, q = 3; with the top row of its coach [1, 5, 3]. 13 is of the form 4q + 1, q = 3; so has a coach of [1, 3, 5]. 37 is of the form 4q + 1, q = 9; so has a coach with the top row composed of a permutation of the first 9 odd integers: [1, 9, 7, 15, 11, 13, 3, 17, 5]. - Gary W. Adamson, Sep 08 2012
Odd primes p such that 2^m is not congruent to 1 or -1 (mod p) for 0 < m < (p-1)/2. - Charles R Greathouse IV, Sep 15 2012
These are also the odd primes a(n) for which there is only one periodic Schick sequence (see the reference, and also the Brändli and Beyne link, eq. (2) for the recurrence but using various inputs. See also a comment in A332439). This sequence has primitive period length (named pes in Schick's book) A003558((a(n)-1)/2) = A005034(a(n)) = A000010(a(n))/2 = (a(n) - 1)/2, for n >= 1. - Wolfdieter Lang, Apr 09 2020
From Jianing Song, Dec 24 2022: (Start)
Primes p such that the multiplicative order of 4 modulo p is (p-1)/2. Proof of equivalence: let ord(a,k) be the multiplicative of a modulo k.
If 2^m is not 1 or -1 (mod p) for 0 < m < (p-1)/2, then ord(2,p) is either p-1 or (p-1)/2. If ord(2,p) = p-1, then ord(4,p) = (p-1)/2. If ord(2,p) = (p-1)/2, then p == 3 (mod 4), otherwise 2^((p-1)/4) == -1 (mod p), so ord(4,p) = (p-1)/2.
Conversely, if ord(4,p) = (p-1)/2, then ord(2,p) = p-1, or ord(2,p) = (p-1)/2 and p == 3 (mod 4) (otherwise ord(4,p) = (p-1)/4). In the first case, (p-1)/2 is the smallest m > 0 such that 2^m == +-1 (mod p); in the second case, since (p-1)/2 is odd, 2^m == -1 (mod p) has no solution. In either case, so 2^m is not 1 or -1 (mod p) for 0 < m < (p-1)/2.
{(a(n)-1)/2} is the sequence of indices of fixed points of A053447.
A prime p is a term if and only if one of the two following conditions holds: (a) 2 is a primitive root modulo p; (b) p == 3 (mod 4), and the multiplicative order of 2 modulo p is (p-1)/2 (in this case, we have p == 7 (mod 8) since 2 is a quadratic residue modulo p). (End)
From Jianing Song, Aug 11 2023: (Start)
Primes p such that 2 or -2 (or both) is a primitive root modulo p. Proof of equivalence: if ord(2,p) = p-1, then clearly ord(4,p) = (p-1)/2. If ord(-2,p) = p-1, then we also have ord(4,p) = (p-1)/2. Conversely, suppose that ord(4,p) = (p-1)/2, then ord(2,p) = p-1 or (p-1)/2, and ord(-2,p) = p-1 or (p-1)/2. If ord(2,p) = ord(-2,p) = (p-1)/2, then we have that (p-1)/2 is odd and (-1)^((p-1)/2) == 1 (mod p), a contradiction.
A prime p is a term if and only if one of the two following conditions holds: (a) -2 is a primitive root modulo p; (b) p == 3 (mod 4), and the multiplicative order of -2 modulo p is (p-1)/2 (in this case, we have p == 3 (mod 8) since -2 is a quadratic residue modulo p). (End)
No terms are congruent to 1 modulo 8, since otherwise we would have 4^((p-1)/4) = (+-2)^((p-1)/2) == 1 (mod p). - Jianing Song, May 14 2024
The n-th prime A000040(n) is a term iff A376010(n) = 2. - Max Alekseyev, Sep 05 2024

			Prime 23 has a k value of 11 = (23 - 1)/2 (Cf. A003558(11)). It follows that 23 has only one coach (A135303(11) = 1). 23 is thus in the set. On the other hand 31 is not in the set since A135303(15) shows 3 coaches, with A003558(15) = 5.
13 is in the set since A135303(6) = 1; but 17 isn't since A135303(8) = 2.

P. Hilton and J. Pedersen, A Mathematical Tapestry, Demonstrating the Beautiful Unity of Mathematics, 2010, Cambridge University Press, pages 260-264.
Carl Schick, Trigonometrie und unterhaltsame Zahlentheorie, Bokos Druck, Zürich, 2003 (ISBN 3-9522917-0-6). Tables 3.1 to 3.10, for odd p = 3..113 (with gaps), pp. 158-166.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Gerold Brändli and Tim Beyne, Modified Congruence Modulo n with Half the Amount of Residues, arXiv:1504.02757 [math.NT], 2016.
Marcelo E. Coniglio, Francesc Esteva, Tommaso Flaminio, and Lluis Godo, On the expressive power of Lukasiewicz's square operator, arXiv:2103.07548 [math.LO], 2021.

Union of A001122 and A105874.
A105876 is the subsequence of terms congruent to 3 modulo 4.
Complement of A268923 in the set of odd primes.
Cf. A082654 (order of 4 mod n-th prime), A000010, A000040, A003558, A005034, A053447, A054639, A135303, A364867, A376010.

isA216371 := proc(n)
    if isprime(n) then
        if A135303((n-1)/2) = 1 then
            true;
        else
            false;
        end if;
    else
        false;
    end if;
end proc:
A216371 := proc(n)
    local p;
    if n = 1 then
        3;
    else
        p := nextprime(procname(n-1)) ;
        while true do
            if isA216371(p) then
                return p;
            end if;
            p := nextprime(p) ;
        end do:
    end if;
end proc:
seq(A216371(n),n=1..40) ; # R. J. Mathar, Dec 01 2014

Suborder[a_, n_] := If[n > 1 && GCD[a, n] == 1, Min[MultiplicativeOrder[a, n, {-1, 1}]], 0]; nn = 150; Select[Prime[Range[2, nn]], EulerPhi[#]/(2*Suborder[2, #]) == 1 &] (* T. D. Noe, Sep 18 2012 *)
f[p_] := Sum[Cos[2^n Pi/((2 p + 1))], {n, p}]; 1 + 2 * Select[Range[500], Reduce[f[#] == -1/2, Rationals] &]; (* Gerry Martens, May 01 2016 *)

is(p)=for(m=1,p\2-1, if(abs(centerlift(Mod(2,p)^m))==1, return(0))); p>2 && isprime(p) \\ Charles R Greathouse IV, Sep 18 2012

is(p) = isprime(p) && (p>2) && znorder(Mod(4,p)) == (p-1)/2 \\ Jianing Song, Dec 24 2022

a(n) = 2*A054639(n) + 1. - L. Edson Jeffery, Dec 18 2012

A136415 Numbers n such that a type-3 Gaussian normal basis over GF(2^n) exists.

Original entry on oeis.org

4, 6, 12, 14, 20, 22, 46, 52, 54, 60, 70, 76, 92, 94, 116, 124, 126, 140, 166, 174, 180, 182, 204, 206, 214, 220, 230, 236, 244, 252, 262, 276, 284, 286, 292, 294, 302, 332, 340, 350, 356, 364, 372, 374, 390, 404, 412, 430, 460, 484, 494, 510, 516, 526, 532

Offset: 1

Joerg Arndt, Mar 31 2008

nonn

A type-t Gaussian normal basis exists for GF(2^n) if p=n*t+1 is prime and gcd(n,(p-1)/ord(2 mod p))==1.

Type-1 basis correspond to sequence A071642, type-2 basis to A054639.

			12 is in the list because 3*12+1=37 is prime and the index of 2 mod 37 (==36/ord(2 mod 37)==1, 2 is a generator mod 37) is coprime to 12.

Joerg Arndt, Mar 31 2008, Table of n, a(n) for n = 1..201
Joerg Arndt, Matters Computational (The Fxtbook), section 42.9 "Gaussian normal bases", pp.914-920

Cf. A071642, A054639.

gauss_test(n, t)=
{ /* test whether a type-t Gaussian normal basis exists for GF(2^n) */
  local( p, r2, g, d );
  p = t*n + 1;
  if ( !isprime(p), return( 0 ) );
  if ( p<=2, return( 0 ) );
  r2 = znorder( Mod(2, p) );
  d = (p-1)/r2;
  g = gcd(d, n);
  return ( if ( 1==g, 1, 0) );
}
/* generate this sequence: */
t=3;ct=1;for(n=1,10^7,if(gauss_test(n,t), print1(n,", ");ct+=1;if(ct>200,break())))

A163779 Numbers k of the form 4j + 1 such that 2k + 1 is a prime with primitive root 2.

Original entry on oeis.org

1, 5, 9, 29, 33, 41, 53, 65, 69, 81, 89, 105, 113, 173, 189, 209, 221, 233, 245, 261, 273, 281, 293, 309, 329, 393, 413, 429, 441, 453, 473, 509, 545, 561, 585, 593, 629, 641, 645, 653, 713, 725, 741, 749, 761, 765, 785, 809, 833, 873, 893, 933, 953, 965, 989, 993

Offset: 1

Peter R. J. Asveld, Aug 12 2009

nonn

Previous name was: a(n) is the n-th A^+_1-prime (Archimedes^+_1 prime).

N is A^+_1-prime iff N=1 (mod 4), p=2N+1 is a prime number and +2 generates Z_p^* (the multiplicative group of Z_p) but -2 does not.

Joerg Arndt, Table of n, a(n) for n = 1..10000 (first 3328 from P. R. J. Asveld)
P. R. J. Asveld, Permuting operations on strings and their relation to prime numbers, Discrete Applied Mathematics 159 (2011) 1915-1932.
P. R. J. Asveld, Permuting operations on strings and the distribution of their prime numbers (2011), TR-CTIT-11-24, Dept. of CS, Twente University of Technology, Enschede, The Netherlands.
P. R. J. Asveld, Some Families of Permutations and Their Primes (2009), TR-CTIT-09-27, Dept. of CS, Twente University of Technology, Enschede, The Netherlands.
P. R. J. Asveld, Permuting Operations on Strings-Their Permutations and Their Primes, Twente University of Technology, 2014.
P. Michael Hutchins, Reworded Definition

The A^+_1-primes are the T- or Twist-primes congruent 1 (mod 4), these T-primes are equal to the Queneau-numbers (A054639). For the related A_0-, A_1- and A^-_1-primes, see A163777, A163778 and A163780. Considered as sets the union of A163779 and A163780 equals A163778, the union of A163779 and A163777 is equal to A163782 (J_2-primes).

okQ[n_] := Mod[n, 4] == 1 && PrimeQ[2n+1] && MultiplicativeOrder[2, 2n+1] == 2n;
Select[Range[1000], okQ] (* Jean-François Alcover, Jun 30 2018, after Andrew Howroyd *)

ok(n) = n%4==1 && isprime(2*n+1) && znorder(Mod(2, 2*n+1))==2*n;
select(ok, [1..1000]) \\ Andrew Howroyd, Nov 11 2017

2 * a(n) + 1 = A213051(n+1). - Joerg Arndt, Mar 23 2018

a(32)-a(55) from Andrew Howroyd, Nov 11 2017

Term 1 prepended and new name from Joerg Arndt, Mar 23 2018

A163781 a(n) is the n-th dJ_2 prime (dual Josephus_2 prime).

Original entry on oeis.org

2, 3, 6, 11, 14, 18, 23, 26, 30, 35, 39, 50, 51, 74, 83, 86, 90, 95, 98, 99, 119, 131, 134, 135, 146, 155, 158, 174, 179, 183, 186, 191, 194, 210, 230, 231, 239, 243, 251, 254, 270, 278, 299, 303, 306, 323, 326, 330, 338, 350, 354, 359, 371, 375, 378

Offset: 1

Peter R. J. Asveld, Aug 17 2009

nonn

The family of dual Josephus_2 (or dJ_2) permutations is defined by p(m,N)=(2N + 1 - F(m,2N + 1))/2 if 1<=m<=N, N>=2, where F(x,y) is the odd number such that 1<=F(x,y)=0. Note that F(2k + 1,y)=2k + 1 for 2k + 1

dJ_2 permutations can also be defined using a numbering/elimination procedure similar to the definition of the Josephus_2 permutations in [R.L. Graham et al.], or in A163782; see [P. R. J. Asveld].

No formula is known for a(n): the dJ_2 primes have been found by exhaustive search. But we have: (1) N is dJ_2 prime iff p=2N+1 is a prime number and -2 generates Z_p^* (the multiplicative group of Z_p); (2) N is dJ_2 prime iff p=2N+1 is a prime number and exactly one of the following holds:

(a) N=2 (mod 4) and both +2 and -2 generate Z_p^*,

(b) N=3 (mod 4) and -2 generates Z_p^* but +2 does not.

			For N=6 we have
  m       | 1   2   3   4   5   6
  --------+----------------------
  F(m,13) | 1   7   3  11   5   9
  t       | 0   2   0   1   0   3
  p(m,6)  | 6   3   5   1   4   2
So the permutation is (1 6 2 3 5 4) and 6 is dJ_2 prime.

R. L. Graham, D. E. Knuth & O. Patashnik, Concrete Mathematics (1989), Addison-Wesley, Reading, MA. Sections 1.3 & 3.3.

P. R. J. Asveld, Table of n, a(n) for n=1..6756
P. R. J. Asveld, Permuting operations on strings and their relation to prime numbers, Discrete Applied Mathematics 159 (2011) 1915-1932.
P. R. J. Asveld, Permuting operations on strings and the distribution of their prime numbers (2011), TR-CTIT-11-24, Dept. of CS, Twente University of Technology, Enschede, The Netherlands.
P. R. J. Asveld, Some Families of Permutations and Their Primes (2009), TR-CTIT-09-27, Dept. of CS, Twente University of Technology, Enschede, The Netherlands.
P. R. J. Asveld, Permuting Operations on Strings-Their Permutations and Their Primes, Twente University of Technology, 2014. University link.
Index entries for sequences related to the Josephus Problem

Considered as sets the union of A163781 and A163782 (J_2 primes) equals A054639 (T-primes or Queneau numbers), their intersection is equal to A163777 (Archimedes_0 primes). A163781 equals the union of A163777 and A163780 (Archimedes^-_1 primes).

okQ[n_] := Mod[n, 4] >= 2 && PrimeQ[2n+1] && MultiplicativeOrder[2, 2n+1] == If[OddQ[n], n, 2n];
Select[Range[1000], okQ] (* Jean-François Alcover, Sep 23 2019, from PARI *)

ok(n)={n%4>=2 && isprime(2*n+1) && znorder(Mod(2, 2*n+1)) == if(n%2,n,2*n)};
select(ok, [1..1000]) \\ Andrew Howroyd, Nov 11 2017

a(37)-a(55) from Andrew Howroyd, Nov 11 2017

A220555 T(n,k) = maximal order N of cyclic group {D,D^2,...,D^N} generated by an n X n Danzer matrix D over Z/kZ, where D is from the m-th Danzer basis and m=2*n+1.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 8, 1, 1, 7, 26, 6, 1, 1, 31, 18, 14, 20, 1, 1, 63, 121, 14, 62, 24, 1, 1, 15, 26, 62, 62, 182, 16, 1, 1, 15, 24, 126, 781, 126, 42, 12, 1, 1, 511, 1640, 30, 24, 3751, 114, 28, 24, 1, 1, 63, 9841, 30, 20, 1638, 2801, 28, 78, 60, 1

Offset: 1

L. Edson Jeffery, Dec 15 2012

For definition of Danzer matrix see [Jeffery] (notation differs there!).
Conjecture 1. Let F_n(x)=sum_{j=0..n} A187660(n,j)*x^{(n-1)*j}. Let f_n in Z[x] be any polynomial in x of degree d such that 0<=d<=(n-1)*(n-2). Then the sequence of coefficients of the series expansion of f_n(x)/F_n(x), when taken over Z/kZ, is periodic with period p <= (n-1)*A220555(n,k), for all n,k > 1. (Cf. [Coleman, et al.] for the case for n=2 (generalized Fibonacci).)
Conjecture 2. If G a cyclic multiplicative group generated by an n X n integer matrix over Z/kZ, then |G|<=T(r,k), for some r<=n.
Definition. If T(n,k)>=(k^n-1)/(k-1), for some k>1, then T(n,k) is said to be "optimal."
Conjecture 3. If T(n,k) is optimal, then n is a Queneau number (A054639).
Sequence is read from antidiagonals of array T which begins as
.1...1....1....1......1.......1......1....1.....1.........1
.1...3....8....6.....20......24.....16...12....24........60
.1...7...26...14.....62.....182.....42...28....78.......434
.1...7...18...14.....62.....126....114...28....54.......434
.1..31..121...62....781....3751...2801..124...363.....24211
.1..63...26..126.....24....1638..13072..252....78.......504
.1..15...24...30.....20.....120....400...60....72........60
.1..15.1640...30..32552....4920.240200...60..4920....488280
.1.511.9841.1022.488281.5028751....342.2044.29523.249511591
.1..63...78..126....124....1638.....42..252...234......7812
Rows might be related to Jordan totient functions J_n(k), however, some entries T(n,k) are products of factors of the form (j^n-1)/(j-1).

D. A. Coleman et al., Periods of (q,r)-Fibonacci sequences and Elliptic Curves, Fibonacci Quart. 44, no 1 (2006) 59-70.
L. E. Jeffery, Danzer matrices.

Cf. A001175 (possibly = row 2), A086839 (possibly = column 2), A160893, A160895, A160897, A160960, A160972, A161010, A161025, A161139, A161167, A161213.

Cf. A187772 (gives maximal periods p of Conjecture 1).

Showing 1-10 of 14 results. Next

Previous name was: a(n) is the n-th A_0-prime (Archimedes_0 prime).

We have: (1) N is A_0-prime if and only if N is even, p = 2N + 1 is a prime number and both +2 and -2 generate Z_p^* (the multiplicative group of Z_p); (2) N is A_0-prime if and only if N = 2 (mod 4), p = 2N + 1 is a prime number and both +2 and -2 generate Z_p^*.

Previous name was: The A_1-primes (Archimedes_1 primes).
We have: (1) N is an A_1-prime iff N is odd, p=2N+1 is a prime number and only one of +2 and -2 generates Z_p^* (the multiplicative group of Z_p); (2) N is an A_1-prime iff p=2N+1 is a prime number and exactly one of the following holds: (a) N == 1 (mod 4) and +2 generates Z_p^* but -2 does not, (b) N == 3 (mod 4) and -2 generates Z_p^* but +2 does not.
The A_1-primes are the odd T- or Twist-primes (the T-primes are the same as the Queneau-numbers, A054639). For the related A_0-, A^+_1- and A^-_1-primes, see A163777, A163779 and A163780. Considered as a set, the present sequence is the union of the A^+_1-primes (A163779) and the A^-_1-primes (A163780). It is also equal to the difference of A054639 and the A_0-primes (A163777).

Previous name was: a(n) is the n-th A^-_1-prime (Archimedes^-_1 prime).

N is A^-_1-prime iff N=3 (mod 4), p=2N+1 is a prime number and -2 generates Z_p^* (the multiplicative group of Z_p) but +2 does not.

Place the numbers 1..N (N>=2) on a circle and cyclically mark the 2nd unmarked number until all N numbers are marked. The order in which the N numbers are marked defines a permutation; N is a J_2-prime if this permutation consists of a single cycle of length N.
The resulting permutation can be written as p(m,N) = (2N+1-||2N+1-m||)/2 (1 <= m <= N), where ||x|| is the odd number such that x/||x|| is a power of 2. E.g., ||16||=1 and ||120||=15.
No formula is known for a(n): the J_2-primes have been found by exhaustive search (however, see the CROSS-REFERENCES). But we have: (1) N is J_2-prime iff p=2N+1 is a prime number and +2 generates Z_p^* (the multiplicative group of Z_p). (2) N is J_2-prime iff p=2N+1 is a prime number and exactly one of the following holds: (a) N == 1 (mod 4) and +2 generates Z_p^* but -2 does not, (b) N == 2 (mod 4) and both +2 and -2 generate Z_p^*.

cp's OEIS Frontend

A163777 Even terms in the sequence of Queneau numbers A054639.

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A163778 Odd terms in A054639.

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Mathematica

PARI

PARI

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A163780 Terms in A054639 equal to 3 mod 4.

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PARI

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A163782 a(n) is the n-th J_2-prime (Josephus_2 prime).

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Java

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PARI

PARI

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A071642 Numbers n such that x^n + x^(n-1) + x^(n-2) + ... + x + 1 is irreducible over GF(2).

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A216371 Odd primes with one coach: primes p such that A135303((p-1)/2) = 1.

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Maple

Mathematica

PARI

PARI

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A136415 Numbers n such that a type-3 Gaussian normal basis over GF(2^n) exists.

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A163779 Numbers k of the form 4j + 1 such that 2k + 1 is a prime with primitive root 2.