

A163778


a(n) is the nth A_1prime (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
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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): N1 N3 ... 5 3 1 2 4 6 ... N1 N (N is even). N N2 ... 5 3 1 3 4 6 ... N3 N1 (N is odd), N is 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 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 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.


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) 1<=m<=N
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 A_1prime but 7 is not.


CROSSREFS

The A_1primes are the odd T or Twistprimes, these Tprimes are equal to the Queneaunumbers (A054639). For the related A_0, A^+_1 and A^_1primes, see A163777, A163779 and A163780. Considered as sets A163778 is the union of the A^+_1primes (A163779) and the A^_1primes (A163780), it also equals the difference of A054639 and the A_0primes (A163777).
Sequence in context: A113488 A092917 A256220 * A160358 A120806 A020946
Adjacent sequences: A163775 A163776 A163777 * A163779 A163780 A163781


KEYWORD

nonn,more


AUTHOR

Peter R. J. Asveld, Aug 11 2009


STATUS

approved



