A163778 Odd terms in A054639.

3, 5, 9, 11, 23, 29, 33, 35, 39, 41, 51, 53, 65, 69, 81, 83, 89, 95, 99, 105, 113, 119, 131, 135, 155, 173, 179, 183, 189, 191, 209, 221, 231, 233, 239, 243, 245, 251, 261, 273, 281, 293, 299, 303, 309, 323, 329, 359, 371, 375, 393, 411, 413, 419, 429

Offset

1,1

Comments

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).

Links

P. R. J. Asveld, Table of n, a(n) for n=1..6706.

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.

Mathematica

follow[s_, f_] := Module[{t, k}, t = f[s]; k = 1; While[t>s, k++; t = f[t]]; If[s == t, k, 0]];
okQ[n_] := n>1 && n == follow[1, Function[j, Ceiling[n/2] + (-1)^j*Ceiling[ (j-1)/2]]];
A163778 = Select[Range[1000], okQ] (* Jean-François Alcover, Jun 07 2018, after Andrew Howroyd *)

Prog

(PARI)
Follow(s, f)={my(t=f(s), k=1); while(t>s, k++; t=f(t)); if(s==t, k, 0)}
ok(n)={n>1 && n==Follow(1, j->ceil(n/2) + (-1)^j*ceil((j-1)/2))}
select(ok, [1..1000]) \\ Andrew Howroyd, Nov 11 2017
(PARI)
ok(n)={n>1 && n%2==1 && isprime(2*n+1) && znorder(Mod(2, 2*n+1)) == if(n%4==3, n, 2*n)}
select(ok, [1..1000]) \\ Andrew Howroyd, Nov 11 2017

Crossrefs

Cf. A054639, A163777, A163779, A163780, A294434, A294673.

Sequence in context: A113488 A092917 A256220 * A328643 A160358 A319084

Adjacent sequences: A163775 A163776 A163777 * A163779 A163780 A163781

Keyword

nonn

Author

Peter R. J. Asveld, Aug 11 2009

Extensions

a(33)-a(55) from Andrew Howroyd, Nov 11 2017

New name from Joerg Arndt, Mar 23 2018

Status

approved