

A163778


List of the A_1primes (or Archimedes_1 primes).


7



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
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OFFSET

1,1


COMMENTS

Archimedes' spiral with polar equation r=c(pi + a) (c>0; a is the angle) defines a family of permutations: 1 is placed at the origin of the XYplane and each time, as a increases, that r hits the Xaxis we put the next number on the Xaxis; reading the numbers from left to right yields the permutation of 1..N (N>=2) given by N1 N3 ... 5 3 1 2 4 6 ... N1 N (if N is even) or N N2 ... 5 3 1 2 4 6 ... N3 N1 (if N is odd). Then N is an A_1prime if this permutation consists of a single cycle of length N. So all A_1primes are odd.
No formula is known for a(n): the A_1primes have been found by exhaustive search. But we have: (1) N is an A_1prime 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_1prime 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_1primes are the odd T or Twistprimes (the Tprimes are the same as the Queneaunumbers, A054639). For the related A_0, A^+_1 and A^_1primes, see A163777, A163779 and A163780. Considered as a set, the present sequence is the union of the A^+_1primes (A163779) and the A^_1primes (A163780). It is also equal to the difference of A054639 and the A_0primes (A163777).


REFERENCES

P. R. J. Asveld, Permuting Operations on StringsTheir Permutations and Their Primes, Twente University of Technology, 2014; http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.216.1682; http://doc.utwente.nl/67513/1/pospp.pdf.


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) 19151932.
P. R. J. Asveld, Permuting operations on strings and the distribution of their prime numbers (2011), TRCTIT1124, Dept. of CS, Twente University of Technology, Enschede, The Netherlands.
P. R. J. Asveld, Some Families of Permutations and Their Primes (2009), TRCTIT0927, Dept. of CS, Twente University of Technology, Enschede, The Netherlands.


FORMULA

The permutation can be written as
p(m,N) = ceiling(N/2) + (1)^m*ceiling((m1)/2), for 1<=m<=N.
For example, p(1,5)=3, p(2,5)=4, p(3,5)=2, p(4,5)=5, p(5,5)=1


EXAMPLE

For N=5 and N=7 we have (1 3 2 4 5) and (1 4 6 7)(2 5)(3) as permutations, so 5 is an A_1prime but 7 is not.


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((j1)/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, A294673.
Sequence in context: A113488 A092917 A256220 * A160358 A120806 A020946
Adjacent sequences: A163775 A163776 A163777 * A163779 A163780 A163781


KEYWORD

nonn,changed


AUTHOR

Peter R. J. Asveld, Aug 11 2009


EXTENSIONS

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


STATUS

approved



