cp's OEIS Frontend

A106309 Primes p such that for all initial conditions (x(0),x(1),x(2),x(3),x(4)) in [0..p-1]^5 except [0,0,0,0,0], the 5-step recurrence x(k) = x(k-1) + x(k-2) + x(k-3) + x(k-4) + x(k-5) (mod p) has the same period.

%I A106309 #38 Jul 22 2025 13:10:30
%S A106309 5,7,11,13,17,31,37,41,53,79,107,199,233,239,311,331,337,389,463,523,
%T A106309 541,547,557,563,577,677,769,853,937,971,1009,1021,1033,1049,1061,
%U A106309 1201,1237,1291,1307,1361,1427,1453,1543,1657,1699,1723,1747,1753,1759,1787,1801,1811,1861,1877,1997,1999
%N A106309 Primes p such that for all initial conditions (x(0),x(1),x(2),x(3),x(4)) in [0..p-1]^5 except [0,0,0,0,0], the 5-step recurrence x(k) = x(k-1) + x(k-2) + x(k-3) + x(k-4) + x(k-5) (mod p) has the same period.
%C A106309 The first term not in A371566 is a(105) = 4259.
%H A106309 Robert Israel, <a href="/A106309/b106309.txt">Table of n, a(n) for n = 1..10000</a>
%H A106309 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Fibonaccin-StepNumber.html">Fibonacci n-Step Number</a>
%H A106309 Robert Israel, <a href="/A106309/a106309.pdf">Linear Recurrences With a Single Minimal Period</a>
%e A106309 a(3) = 11 is a term because the recurrence has period 16105 for all initial conditions except (0,0,0,0,0).
%p A106309 filter:= proc(p) local Q,q,F,i,z,d,k,kp,G,alpha;
%p A106309   Q:= z^5  - z^4 - z^3 - z^2 - z - 1;
%p A106309   if Irreduc(Q) mod p then return true fi;
%p A106309   F:= (Factors(Q) mod p)[2];
%p A106309   if ormap(t -> t[2]>1, F) then return false fi;
%p A106309   for i from 1 to nops(F) do
%p A106309     q:= F[i][1];
%p A106309     d:= degree(q);
%p A106309     if d = 1 then
%p A106309        kp:= numtheory:-order(solve(q,z),p);
%p A106309     else
%p A106309        G:= GF(p,d, q);
%p A106309        alpha:= G:-ConvertIn(z);
%p A106309        kp:= G:-order(alpha);
%p A106309     fi;
%p A106309     if i = 1 then k:= kp
%p A106309     elif kp <> k then return false
%p A106309     fi;
%p A106309   od;
%p A106309   true
%p A106309 end proc:
%p A106309 select(filter, [seq(ithprime(i),i=1..1000)]);
%Y A106309 Cf. A106287 (orbits of 5-step sequences). Contains A371566.
%K A106309 nonn
%O A106309 1,1
%A A106309 _T. D. Noe_, May 02 2005, revised May 12 2005
%E A106309 4259 found by D. S. McNeil.
%E A106309 Edited by _Robert Israel_, Mar 27 2024