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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.
6
5, 7, 11, 13, 17, 31, 37, 41, 53, 79, 107, 199, 233, 239, 311, 331, 337, 389, 463, 523, 541, 547, 557, 563, 577, 677, 769, 853, 937, 971, 1009, 1021, 1033, 1049, 1061, 1201, 1237, 1291, 1307, 1361, 1427, 1453, 1543, 1657, 1699, 1723, 1747, 1753, 1759, 1787, 1801, 1811, 1861, 1877, 1997, 1999
OFFSET
1,1
COMMENTS
The first term not in A371566 is a(105) = 4259.
LINKS
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number
EXAMPLE
a(3) = 11 is a term because the recurrence has period 16105 for all initial conditions except (0,0,0,0,0).
MAPLE
filter:= proc(p) local Q, q, F, i, z, d, k, kp, G, alpha;
Q:= z^5 - z^4 - z^3 - z^2 - z - 1;
if Irreduc(Q) mod p then return true fi;
F:= (Factors(Q) mod p)[2];
if ormap(t -> t[2]>1, F) then return false fi;
for i from 1 to nops(F) do
q:= F[i][1];
d:= degree(q);
if d = 1 then
kp:= numtheory:-order(solve(q, z), p);
else
G:= GF(p, d, q);
alpha:= G:-ConvertIn(z);
kp:= G:-order(alpha);
fi;
if i = 1 then k:= kp
elif kp <> k then return false
fi;
od;
true
end proc:
select(filter, [seq(ithprime(i), i=1..1000)]);
CROSSREFS
Cf. A106287 (orbits of 5-step sequences). Contains A371566.
Sequence in context: A038958 A109416 A132170 * A371566 A227576 A114262
KEYWORD
nonn,more
AUTHOR
T. D. Noe, May 02 2005, revised May 12 2005
EXTENSIONS
4259 found by D. S. McNeil.
Edited by Robert Israel, Mar 27 2024
STATUS
approved