A106306 Primes that yield a simple orbit structure in 2-step recursions.

2, 3, 7, 13, 17, 23, 37, 41, 43, 47, 53, 61, 67, 73, 83, 89, 97, 103, 107, 109, 113, 127, 137, 149, 157, 163, 167, 173, 193, 197, 223, 227, 233, 241, 257, 263, 269, 277, 281, 283, 293, 307, 313, 317, 337, 347, 353, 367, 373, 383, 389, 397, 401, 409, 421, 433

Offset

1,1

Comments

Consider the 2-step recursion x(k)=x(k-1)+x(k-2) mod n. For any of the n^2 initial conditions x(1) and x(2) in Zn, the recursion has a finite period. When n is a prime in this sequence, all of the orbits, except the one containing (0,0), have the same length.

Except for 5, this appears to be the complement of A053032, odd primes p with one 0 in Fibonacci numbers mod p. - T. D. Noe, May 03 2005

A prime p is in this sequence if either (1) the polynomial x^2-x-1 mod p has no zeros for x in [0,p-1] (see A086937) or (2) the polynomial has zeros, but none is a root of unity mod p. The first few primes in the second category are 41, 61, 89 and 109. - T. D. Noe, May 12 2005

Links

Table of n, a(n) for n=1..56.

Eric Weisstein's World of Mathematics, Fibonacci n-Step Number.

Crossrefs

Cf. A015134 (orbits of 2-step sequences).

Sequence in context: A045328 A045329 A271666 * A069104 A003631 A175443

Adjacent sequences: A106303 A106304 A106305 * A106307 A106308 A106309

Keyword

nonn

Author

T. D. Noe, May 02 2005

Status

approved