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A163778 a(n) is the n-th A_1-prime (Archimedes_1 prime). 6
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 (list; graph; refs; listen; history; text; internal format)



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): N-1 N-3 ... 5 3 1 2 4 6 ... N-1 N (N is even). N N-2 ... 5 3 1 3 4 6 ... N-3 N-1 (N is odd), N is 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 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 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.


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=; http://doc.utwente.nl/67513/1/pospp.pdf.


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.


The permutation can be written as

p(m,N)=ceiling(N/2) + (-1)^m.ceiling((m-1)/2) 1<=m<=N

p(1,5)=3, p(2,5)=4, p(3,5)=2, p(4,5)=5, p(5,5)=1


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 A_1-prime but 7 is not.


The A_1-primes are the odd T- or Twist-primes, these T-primes are equal to the Queneau-numbers (A054639). For the related A_0-, A^+_1- and A^-_1-primes, see A163777, A163779 and A163780. Considered as sets A163778 is the union of the A^+_1-primes (A163779) and the A^-_1-primes (A163780), it also equals the difference of A054639 and the A_0-primes (A163777).

Sequence in context: A113488 A092917 A256220 * A160358 A120806 A020946

Adjacent sequences:  A163775 A163776 A163777 * A163779 A163780 A163781




Peter R. J. Asveld, Aug 11 2009



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Last modified September 25 19:48 EDT 2017. Contains 292500 sequences.