OFFSET
1,1
COMMENTS
Place the numbers 1..N (N>=2) on a circle and cyclically mark the 2nd unmarked number until all N numbers are marked. The order in which the N numbers are marked defines a permutation; N is a J_2-prime if this permutation consists of a single cycle of length N.
The resulting permutation can be written as p(m,N) = (2N+1-||_2N+1-m_||)/2 (1 <= m <= N), where ||_x_|| is the odd number such that x/||_x_|| is a power of 2. E.g., ||_16_||=1 and ||_120_||=15.
No formula is known for a(n): the J_2-primes have been found by exhaustive search (however, see the CROSS-REFERENCES). But we have: (1) N is J_2-prime iff p=2N+1 is a prime number and +2 generates Z_p^* (the multiplicative group of Z_p). (2) N is J_2-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 == 2 (mod 4) and both +2 and -2 generate Z_p^*.
REFERENCES
R. L. Graham, D. E. Knuth & O. Patashnik, Concrete Mathematics (1989), Addison-Wesley, Reading, MA. Sections 1.3 & 3.3.
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; alternative link.
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.
Eric Weisstein's World of Mathematics, Josephus Problem
Wikipedia, Josephus Problem
FORMULA
a(n) = A071642(n+3)/2.
EXAMPLE
p(1,5)=3, p(2,5)=1, p(3,5)=5, p(4,5)=2 and p(5,5)=4.
So p=(1 3 5 4 2) and 5 is J_2-prime.
MATHEMATICA
lst = {};
Do[If[IntegerQ[(2^n + 1)/(2 n + 1)] && PrimitiveRoot[2 n + 1] == 2,
AppendTo[lst, n]], {n, 2, 10^5}]; lst (* Hilko Koning, Sep 21 2021 *)
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)={my(d=2*n+1); n>1&&n==Follow(1, i->(d-((d-i)>>valuation(d-i, 2)))/2)}
select(n->ok(n), [1..1000]) \\ Andrew Howroyd, Nov 11 2017
(PARI)
forprime(p=5, 2000, if(znorder(Mod(2, p))==p-1, print1((p-1)/2, ", "))); \\ Andrew Howroyd, Nov 11 2017
(Java)
isJ2Prime(int n) { // for n > 1
int count = 0, leader = 0;
if (n % 4 == 1 || n % 4 == 2) { // small optimization
do {
leader = A025480(leader + n);
count++;
} while (leader != 0);
}
return count == n;
} // Joe Nellis, Jan 27 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Peter R. J. Asveld, Aug 05 2009
STATUS
approved