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A106307 Primes that yield a simple orbit structure in 3-step recursions. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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

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Last modified April 15 09:32 EDT 2021. Contains 342977 sequences. (Running on oeis4.)