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%I #55 Jun 07 2018 09:51:18
%S 3,5,9,11,23,29,33,35,39,41,51,53,65,69,81,83,89,95,99,105,113,119,
%T 131,135,155,173,179,183,189,191,209,221,231,233,239,243,245,251,261,
%U 273,281,293,299,303,309,323,329,359,371,375,393,411,413,419,429
%N Odd terms in A054639.
%C Previous name was: The A_1-primes (Archimedes_1 primes).
%C 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.
%C 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).
%H P. R. J. Asveld, <a href="/A163778/b163778.txt">Table of n, a(n) for n=1..6706</a>.
%H P. R. J. Asveld, <a href="http://dx.doi.org/10.1016/j.dam.2011.07.019">Permuting operations on strings and their relation to prime numbers</a>, Discrete Applied Mathematics 159 (2011) 1915-1932.
%H P. R. J. Asveld, <a href="http://eprints.eemcs.utwente.nl/20685/">Permuting operations on strings and the distribution of their prime numbers</a> (2011), TR-CTIT-11-24, Dept. of CS, Twente University of Technology, Enschede, The Netherlands.
%H P. R. J. Asveld, <a href="http://eprints.eemcs.utwente.nl/15678/">Some Families of Permutations and Their Primes </a> (2009), TR-CTIT-09-27, Dept. of CS, Twente University of Technology, Enschede, The Netherlands.
%H P. R. J. Asveld, <a href="http://doc.utwente.nl/67513/">Permuting Operations on Strings-Their Permutations and Their Primes</a>, Twente University of Technology, 2014.
%t follow[s_, f_] := Module[{t, k}, t = f[s]; k = 1; While[t>s, k++; t = f[t]]; If[s == t, k, 0]];
%t okQ[n_] := n>1 && n == follow[1, Function[j, Ceiling[n/2] + (-1)^j*Ceiling[ (j-1)/2]]];
%t A163778 = Select[Range[1000], okQ] (* _Jean-François Alcover_, Jun 07 2018, after _Andrew Howroyd_ *)
%o (PARI)
%o Follow(s, f)={my(t=f(s), k=1); while(t>s, k++; t=f(t)); if(s==t, k, 0)}
%o ok(n)={n>1 && n==Follow(1, j->ceil(n/2) + (-1)^j*ceil((j-1)/2))}
%o select(ok, [1..1000]) \\ _Andrew Howroyd_, Nov 11 2017
%o (PARI)
%o ok(n)={n>1 && n%2==1 && isprime(2*n+1) && znorder(Mod(2, 2*n+1)) == if(n%4==3, n, 2*n)}
%o select(ok, [1..1000]) \\ _Andrew Howroyd_, Nov 11 2017
%Y Cf. A054639, A163777, A163779, A163780, A294434, A294673.
%K nonn
%O 1,1
%A _Peter R. J. Asveld_, Aug 11 2009
%E a(33)-a(55) from _Andrew Howroyd_, Nov 11 2017
%E New name from _Joerg Arndt_, Mar 23 2018
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