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A163778 Odd terms in A054639. 9
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 (list; graph; refs; listen; history; text; internal format)
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 * A160358 A120806 A020946

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

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Last modified July 23 02:49 EDT 2018. Contains 312920 sequences. (Running on oeis4.)