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A163778 List of the A_1-primes (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 (list; graph; refs; listen; history; text; internal format)
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 XY-plane and each time, as a increases, that r hits the X-axis we put the next number on the X-axis; reading the numbers from left to right yields the permutation of 1..N (N>=2) given by N-1 N-3 ... 5 3 1 2 4 6 ... N-1 N (if N is even) or N N-2 ... 5 3 1 2 4 6 ... N-3 N-1 (if N is odd). Then N is an A_1-prime if this permutation consists of a single cycle of length N. So all A_1-primes are odd.

No formula is known for a(n): the A_1-primes have been found by exhaustive search. But 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).

REFERENCES

P. R. J. Asveld, Permuting Operations on Strings-Their 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) 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.

FORMULA

The permutation can be written as

p(m,N) = ceiling(N/2) + (-1)^m*ceiling((m-1)/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_1-prime 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((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, 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

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Last modified November 18 08:47 EST 2017. Contains 294861 sequences.