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A106304 Period of the Fibonacci 5-step sequence A001591 mod prime(n).

%I A106304 #28 Feb 16 2025 08:32:57
%S A106304 6,104,781,2801,16105,30941,88741,13032,12166,70728,190861,1926221,
%T A106304 2896405,79506,736,8042221,102689,3720,20151120,2863280,546120,
%U A106304 39449441,48030024,3690720,29509760,104060400,37516960,132316201,28231632,6384,86714880,2248090,3128
%N A106304 Period of the Fibonacci 5-step sequence A001591 mod prime(n).
%C A106304 This sequence is the same as the period of the Lucas 5-step sequence (A106298) mod prime(n) except for n=1 and 109, which correspond to the primes 2 and 599, because 9584 is the discriminant of the characteristic polynomial x^5-x^4-x^3-x^2-x-1 and the prime factors of 9584 are 2 and 599. We have a(n) < prime(n) for the primes in A106281.
%H A106304 Chai Wah Wu, <a href="/A106304/b106304.txt">Table of n, a(n) for n = 1..86</a>
%H A106304 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Fibonaccin-StepNumber.html">Fibonacci n-Step Number</a>
%F A106304 a(n) = A106303(prime(n)).
%t A106304 n=5; Table[p=Prime[i]; a=Join[Table[ -1, {n-1}], {n}]; a=Mod[a, p]; a0=a; k=0; While[k++; s=Mod[Plus@@a, p]; a=RotateLeft[a]; a[[n]]=s; a!=a0]; k, {i, 40}]
%o A106304 (Python)
%o A106304 from itertools import count
%o A106304 from sympy import prime
%o A106304 def A106304(n):
%o A106304     a = b = (0,)*4+(1 % (p:= prime(n)),)
%o A106304     s = 1 % p
%o A106304     for m in count(1):
%o A106304         b, s = b[1:] + (s,), (s+s-b[0]) % p
%o A106304         if a == b:
%o A106304             return m # _Chai Wah Wu_, Feb 22-27 2022
%Y A106304 Cf. A106281 (primes p such that x^5-x^4-x^3-x^2-x-1 mod p has 5 distinct zeros).
%K A106304 nonn
%O A106304 1,1
%A A106304 _T. D. Noe_, May 02 2005, Nov 19 2006
%E A106304 a(31)-a(33) from _Chai Wah Wu_, Feb 27 2022