A106307 - OEIS

%I #10 Mar 24 2024 07:58:39

%S 3,5,23,31,37,59,67,71,89,97,103,113,137,157,179,181,191,223,229,251,

%T 313,317,331,353,367,379,383,389,433,443,449,463,467,487,509,521,577,

%U 587,619,631,641,643,647,653,661,691,709,719,727,751,797,823,829

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

%C 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.

%C 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.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Fibonaccin-StepNumber.html">Fibonacci n-Step Number</a>.

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

%K nonn

%O 1,1

%A _T. D. Noe_, May 02 2005, revised May 12 2005