A106307 Primes that yield a simple orbit structure in 3-step recursions.

3, 5, 23, 31, 37, 59, 67, 71, 89, 97, 103, 113, 137, 157, 179, 181, 191, 223, 229, 251, 313, 317, 331, 353, 367, 379, 383, 389, 433, 443, 449, 463, 467, 487, 509, 521, 577, 587, 619, 631, 641, 643, 647, 653, 661, 691, 709, 719, 727, 751, 797, 823, 829

Offset

1,1

Comments

Consider the 3-step recursion x(k)=x(k-1)+x(k-2)+x(k-3) mod n. For any of the n^3 initial conditions x(1), x(2) and x(3) in Zn, the recursion has a finite period. When n is a prime in this sequence, all of the orbits, except the one containing (0,0,0), have the same length.

A prime p is in this sequence if either (1) the polynomial x^3-x^2-x-1 mod p has no zeros for x in [0,p-1] (see A106282) or (2) the polynomial has zeros, but none is a root of unity mod p. The first two primes in the second category are 103 and 587.

Links

Table of n, a(n) for n=1..53.

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

Crossrefs

Cf. A106285 (orbits of 3-step sequences).

Sequence in context: A136891 A296920 A106857 * A106282 A163153 A339414

Adjacent sequences: A106304 A106305 A106306 * A106308 A106309 A106310

Keyword

nonn

Author

T. D. Noe, May 02 2005, revised May 12 2005

Status

approved