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A106306 Primes that yield a simple orbit structure in 2-step recursions. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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

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Last modified August 3 05:55 EDT 2020. Contains 336197 sequences. (Running on oeis4.)